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Tense MV-algebras and related functions Jan Paseka Michal Botur - PowerPoint PPT Presentation

Nov 16, 2022 •370 likes •1.59k views

Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Tense MV-algebras

  1. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  2. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  3. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  4. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-algebras Given a positive integer n ∈ N , we let nx = x ⊕ x ⊕ x ···⊕ x , n times, x n = x ⊙ x ⊙ x ···⊙ x , n times, 0 x = 0 and x 0 = 1 . In every MV-algebra M the following equalities hold (whenever the respective join or meet exist): (i) a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , a ⊕ � i ∈ I x i = � i ∈ I ( a ⊕ x i ) , (ii) a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , a ⊙ � i ∈ I x i = � i ∈ I ( a ⊙ x i ) , Operators on MV-algebras 8 / 42

  5. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  6. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  7. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – MV-morphisms and filters Morphisms of MV-algebras (shortly MV-morphisms ) are defined as usual, they are functions which preserve the binary operations ⊕ and ⊙ , the unary operation ¬ and nullary operations 0 and 1 . A filter of a MV-algebra M is a subset F ⊆ M satisfying: (F1) 1 ∈ F (F2) x ∈ F , y ∈ M , x ≤ y ⇒ y ∈ F (F3) x , y ∈ F ⇒ x ⊙ y ∈ F . A filter is said to be proper if 0 �∈ F . Note that there is one-to-one correspondence between filters and congruences on MV-algebras. Operators on MV-algebras 9 / 42

  8. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  9. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  10. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Prime and maximal filters A filter Q is prime if it satisfies the following conditions: (P1) 0 / ∈ Q . (P2) For each x , y in M such that x ∨ y ∈ Q , either x ∈ Q or y ∈ Q . In this case the corresponding factor MV-algebra M / Q is linear. A filter U is maximal (and in this case it will be also called an ultrafilter) if 0 / ∈ U and for any other filter F of M such that U ⊆ F , then either F = M or F = U . There is a one-to-one correspondence between ultrafilters and MV-morphisms from M into [ 0 , 1 ] . An MV-algebra M is called semisimple if the intersection of all its maximal filters is { 1 } . Operators on MV-algebras 10 / 42

  11. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  12. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  13. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  14. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Basic definitions – Boolean elements and states An element a of a MV-algebra M is said to be Boolean if a ⊕ a = a . We say that a MV-algebra M is Boolean if every element of M is Boolean. For a MV-algebra M , the set B ( M ) of all Boolean elements is a Boolean algebra. We say that a state on a MV-algebra M is any mapping s : M → [ 0 , 1 ] such that (i) s ( 1 ) = 1 , and (ii) s ( a ⊕ b ) = s ( a )+ s ( b ) whenever a ⊙ b = 0 . A state s is extremal if, for all states s 1 , s 2 such that s = λ s 1 +( 1 − λ ) s 2 for λ ∈ ( 0 , 1 ) we conclude s = s 1 = s 2 . We recall that a state s is extremal iff { a ∈ M : s ( a ) = 1 } is an ultrafilter of M iff s ( a ⊕ b ) = min { s ( a )+ s ( b ) , 1 } , a , b ∈ M iff s is a morphism of MV-algebras. Operators on MV-algebras 11 / 42

  15. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 12 / 42

  16. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Dyadic numbers and MV-terms The set D of dyadic numbers is the set of the rational numbers that can be written as a finite sum of power of 2. If a is a number of [ 0 , 1 ] , a dyadic decomposition of a is a sequence a ∗ = ( a i ) i ∈ N of elements of { 0 , 1 } such that a = ∑ ∞ i = 1 a i 2 − i . We denote by a ∗ i the i th element of any sequence (of length greater than i ) a ∗ . If a is a dyadic number of [ 0 , 1 ] , then a admits a unique finite dyadic decomposition, called the dyadic decomposition of a . If a ∗ is a dyadic decomposition of a real a and if k is a positive integer then we denote by � a ∗ � k the finite sequence ( a 1 ,..., a k ) defined by the first k elements of a ∗ and by � a ∗ � k the dyadic number ∑ k i = 1 a i 2 − i . Operators on MV-algebras 13 / 42

  17. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Dyadic numbers and MV-terms Definition (Ostermann,Teheux) We denote by f 0 ( x ) and f 1 ( x ) the terms x ⊕ x and x ⊙ x respectively, and by T D the clone generated by f 0 ( x ) and f 1 ( x ) . We also denote by g . the mapping between the set of finite sequences of elements of { 0 , 1 } (and thus of dyadic numbers in [ 0 , 1 ] ) and T D defined by: g ( a 1 ,..., a k ) = f a k ◦···◦ f a 1 i = 1 a i 2 − i , we for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } . If a = ∑ k sometimes write g a instead of g ( a 1 ,..., a k ) . We also denote, for a dyadic number a ∈ D ∩ [ 0 , 1 ) and a positive integer k ∈ N such that 2 − k ≤ 1 − a , by l ( a , k ) : [ a , a + 2 − k ] → [ 0 , 1 ] a linear function defined as follows l ( a , k )( x ) = 2 k ( x − a ) for all x ∈ [ a , a + 2 − k ] . Operators on MV-algebras 14 / 42

  18. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  19. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  20. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications MV-terms on the interval [ 0 , 1 ] Lemma (Teheux) If a ∗ = ( a i ) i ∈ N and x ∗ = ( x i ) i ∈ N are dyadic decompositions of two elements of a , x ∈ [ 0 , 1 ] , then, for any positive integer k ∈ N , i = 1 a i 2 − i + 2 − k if x > ∑ k  1  if x < ∑ k i = 1 a i 2 − i g � a ∗ � k ( x ) = 0 l ( � a ∗ � k , k )( x ) = ∑ ∞ i = 1 x i + k 2 − i otherwise.  Note that for any finite sequence ( a 1 ,..., a k ) of elements of { 0 , 1 } such that a k = 0 we have that g ( a 1 ,..., a k ) = g ( a 1 ,..., a k − 1 ) ⊕ g ( a 1 ,..., a k − 1 ) and clearly any dyadic number a corresponds to such a sequence ( a 1 ,..., a k ) . Corollary (Teheux) Let us have the standard MV-algebra [ 0 , 1 ] , x ∈ [ 0 , 1 ] and r ∈ ( 0 , 1 ) ∩ D . Then there is a term t r in T D such that t r ( x ) = 1 if and only if r ≤ x . Operators on MV-algebras 15 / 42

  21. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions t r on unit interval [ 0 , 1 ] Example ( 0 , 1 ) ( 0 . 5 , 1 ) ( 1 , 1 ) ( 0 , 1 ) ( 0 . 75 , 1 ) ( 1 , 1 ) ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ✁ ✄ ( 0 , 0 ) ( 1 , 0 ) ( 0 , 0 ) ( 0 . 5 , 0 ) ( 1 , 0 ) Function t 1 2 = g 0 Function t 3 4 = g 10 Operators on MV-algebras 16 / 42

  22. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 17 / 42

  23. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Lemma Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M such that s ( x ) = 1 . Then, for any n ∈ N , n > 1 , n × x = 1 . Proposition Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M . Then s ( x ) = 1 iff t r ( x ) = 1 for all r ∈ ( 0 , 1 ) ∩ D . Equivalently, s ( x ) < 1 iff there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) � = 1 . In this case, s ( x ) < r . Operators on MV-algebras 18 / 42

  24. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Lemma Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M such that s ( x ) = 1 . Then, for any n ∈ N , n > 1 , n × x = 1 . Proposition Let M be a linearly ordered MV-algebra, s : M → [ 0 , 1 ] an MV-morphism, x ∈ M . Then s ( x ) = 1 iff t r ( x ) = 1 for all r ∈ ( 0 , 1 ) ∩ D . Equivalently, s ( x ) < 1 iff there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) � = 1 . In this case, s ( x ) < r . Operators on MV-algebras 18 / 42

  25. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Proposition Let M be an MV-algebra, x ∈ M and F be any filter of M . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < 1 if and only if there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) / ∈ F . Corollary Let M be an MV-algebra, x ∈ M and F be any filter of M such that t r ( x ) / ∈ F for some dyadic number r ∈ ( 0 , 1 ) ∩ D . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < r < 1 . Operators on MV-algebras 19 / 42

  26. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Filters, ultrafilters and the term t r Proposition Let M be an MV-algebra, x ∈ M and F be any filter of M . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < 1 if and only if there is a dyadic number r ∈ ( 0 , 1 ) ∩ D such that t r ( x ) / ∈ F . Corollary Let M be an MV-algebra, x ∈ M and F be any filter of M such that t r ( x ) / ∈ F for some dyadic number r ∈ ( 0 , 1 ) ∩ D . Then there is an MV-morphism s : M → [ 0 , 1 ] such that s ( F ) ⊆ { 1 } and s ( x ) < r < 1 . Operators on MV-algebras 19 / 42

  27. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 20 / 42

  28. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  29. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  30. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  31. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  32. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  33. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semistates on MV-algebras In this section we characterize arbitrary meets of MV-morphism into a unit interval as so-called semi-states. Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a semi-state on A if (i) s ( 1 ) = 1 , (ii) x ≤ y implies s ( x ) ≤ s ( y ) , (iii) s ( x ) = 1 and s ( y ) = 1 implies s ( x ⊙ y ) = 1 , (iv) s ( x ) ⊙ s ( x ) = s ( x ⊙ x ) , (v) s ( x ) ⊕ s ( x ) = s ( x ⊕ x ) . Operators on MV-algebras 21 / 42

  34. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  35. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  36. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  37. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  38. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  39. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong semistates on MV-algebras Definition Let A be an MV-algebra. A map s : A → [ 0 , 1 ] is called a strong semi-state on A if it is a semistate such that (vi) s ( x ) ⊙ s ( y ) ≤ s ( x ⊙ y ) , (vii) s ( x ) ⊕ s ( y ) ≤ s ( x ⊕ y ) , (viii) s ( x ∧ y ) = s ( x ) ∧ s ( y ) , (ix) s ( x n ) = s ( x ) n for all n ∈ N , (x) n × s ( x ) = s ( n × x ) for all n ∈ N . Note that any MV-morphism into a unit interval is a strong semi-state. Operators on MV-algebras 22 / 42

  40. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  41. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  42. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Meets of MV-morphism Lemma Let A be an MV-algebra, S a non-empty set of semi-states (strong semi-states) on A . Then the point-wise meet t = � S : A → [ 0 , 1 ] is a semi-state (strong semi-state) on A . Lemma Let A be an MV-algebra, s , t semi-states on A . Then t ≤ s iff t ( x ) = 1 implies s ( x ) = 1 for all x ∈ A . Proposition Let A be an MV-algebra, t a semi-state on A and S t = { s : A → [ 0 , 1 ] | s is an MV-morphism , s ≥ t } . Then t = � S t . Operators on MV-algebras 23 / 42

  43. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  44. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  45. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Any semi-state is strong Corollary Any semi-state on an MV-algebra A is a strong semi-state. Corollary The only semi-state s on an MV-algebra A with s ( 0 ) � = 0 is the constant function s ( x ) = 1 for all x ∈ A . Corollary The only semi-state s on the standard MV-algebra [ 0 , 1 ] with s ( 0 ) = 0 is the identity function. Operators on MV-algebras 24 / 42

  46. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  47. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  48. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  49. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  50. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  51. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The dual version Remark It is transparent that all the preceding notions and results including Proposition 8 can be dualized. In particular, any dual semi-state, i.e., a map s : A → [ 0 , 1 ] satisfying conditions (i),(ii),(iv), (v) and the dual condition (iii)’ s ( x ) = 0 and s ( y ) = 0 implies s ( x ⊕ y ) = 0 is a join of extremal states on A . Proposition Let A be an MV-algebra, s a state on A . Then the following conditions are equivalent: (a) s is a morphism of MV-algebras, (b) s satisfies the condition s ( x ∧ x ′ ) = s ( x ) ∧ s ( x ) ′ for all x ∈ A ., (c) s satisfies the condition (iv) from the definition of a semi-state, (d) s satisfies the condition (viii) from the definition of a strong semi-state. Operators on MV-algebras 25 / 42

  52. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 26 / 42

  53. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  54. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  55. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  56. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  57. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  58. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Functions between MV-algebras This section studies the notion of an fm-function between MV-algebras (strong fm-function between MV-algebras). The main purpose of this section is to establish in some sense a canonical construction of strong fm-function between MV-algebras. This construction is an ultimate source of numerous examples. Definition By an fm-function between MV-algebras G is meant a function G : A 1 → A 2 such that A 1 = ( A 1 ; ⊕ 1 , ⊙ 1 , ¬ 1 , 0 1 , 1 1 ) and A 2 = ( A 2 ; ⊕ 2 , ⊙ 2 , ¬ 2 , 0 2 , 1 2 ) are MV-algebras and (FM1) G ( 1 1 ) = 1 2 , (FM2) x ≤ 1 y implies G ( x ) ≤ 2 G ( y ) , (FM3) G ( x ) = 1 2 = G ( y ) implies G ( x ⊙ 1 y ) = 1 2 , (FM4) G ( x ) ⊙ 2 G ( x ) = G ( x ⊙ 1 x ) , (FM5) G ( x ) ⊕ 2 G ( x ) = G ( x ⊕ 1 x ) . Operators on MV-algebras 27 / 42

  59. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  60. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  61. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  62. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  63. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  64. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  65. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  66. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  67. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Definition If a function G between MV-algebras satisfies conditions (FM6) G ( x ) ⊙ 2 G ( y ) ≤ G ( x ⊙ 1 y ) , (FM7) G ( x ) ⊕ 2 G ( y ) ≤ G ( x ⊕ 1 y ) , (FM8) G ( x ) ∧ 2 G ( y ) = G ( x ∧ 1 y ) , (FM9) G ( x n ) = G ( x ) n for all n ∈ N , (FM10) n × 2 G ( x ) = G ( n × 1 x ) for all n ∈ N , we say that G is a strong fm-function between MV-algebras . If G : A 1 → A 2 and H : B 1 → B 2 are fm-functions between MV-algebras, then a morphism between G and H is a pair ( ϕ , ψ ) of morphism of MV-algebras ϕ : A 1 → B 1 and ψ : A 2 → B 2 such that ψ ( G ( x )) = H ( ϕ ( x )) , for any x ∈ A 1 . Operators on MV-algebras 28 / 42

  68. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  69. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  70. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  71. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Strong functions between MV-algebras Note that (FM8) yields (FM2), (FM9) yields (FM4) and (FM10) yields (FM5). Also, a composition of fm-functions (strong fm-functions) is an fm-function (a strong fm-function) again and any morphism of MV-algebras is an fm-function (a strong fm-function). The notion of an fm-function generalizes both the notions of a semi-state and of a ⊙ -operator which is an fm-function G from A 1 to itself such that (FM6) is satisfied. According to both (FM4) and (FM5), G | B ( A 1 ) : B ( A 1 ) → B ( A 2 ) is an fm-function (a strong fm-function) whenever G has the respective property. Lemma Let G : A 1 → A 2 be an fm-function between MV-algebras, r ∈ ( 0 , 1 ) ∩ D . Then t r ( G ( x )) = G ( t r ( x )) for all x ∈ A 1 . Operators on MV-algebras 29 / 42

  72. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras I By a frame is meant a triple ( S , T , R ) where S , T are non-void sets and R ⊆ S × T . Having an MV-algebra M = ( M ; ⊕ , ⊙ , ¬ , 0 , 1 ) and a non-void set T , we can produce the direct power M T = ( M T ; ⊕ , ⊙ , ¬ , o , j ) where the operations ⊕ , ⊙ and ¬ are defined and evaluated on p , q ∈ M T componentwise. Moreover, o , j are such elements of M T that o ( t ) = 0 and j ( t ) = 1 for all t ∈ T . The direct power M T is again an MV-algebra. Operators on MV-algebras 30 / 42

  73. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras I By a frame is meant a triple ( S , T , R ) where S , T are non-void sets and R ⊆ S × T . Having an MV-algebra M = ( M ; ⊕ , ⊙ , ¬ , 0 , 1 ) and a non-void set T , we can produce the direct power M T = ( M T ; ⊕ , ⊙ , ¬ , o , j ) where the operations ⊕ , ⊙ and ¬ are defined and evaluated on p , q ∈ M T componentwise. Moreover, o , j are such elements of M T that o ( t ) = 0 and j ( t ) = 1 for all t ∈ T . The direct power M T is again an MV-algebra. Operators on MV-algebras 30 / 42

  74. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras II Theorem Let M be a linearly ordered complete MV-algebra, ( S , T , R ) be a frame and G ∗ be a map from M T into M S defined by G ∗ ( p )( s ) = � { p ( t ) | t ∈ T , sRt } , for all p ∈ M T and s ∈ S . Then G ∗ is a strong fm-function between MV-algebras which has a left adjoint P ∗ . In this case, for all q ∈ M S and t ∈ T , P ∗ ( q )( t ) = � { q ( s ) | s ∈ T , sRt } and P ∗ : ( M S ) op → ( M T ) op is a strong fm-function between MV-algebras. We say that G ∗ : M T → M S is the canonical strong fm-function between MV-algebras induced by the frame ( S , T , R ) and the MV-algebra M . Operators on MV-algebras 31 / 42

  75. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The construction of strong functions between MV-algebras II Theorem Let M be a linearly ordered complete MV-algebra, ( S , T , R ) be a frame and G ∗ be a map from M T into M S defined by G ∗ ( p )( s ) = � { p ( t ) | t ∈ T , sRt } , for all p ∈ M T and s ∈ S . Then G ∗ is a strong fm-function between MV-algebras which has a left adjoint P ∗ . In this case, for all q ∈ M S and t ∈ T , P ∗ ( q )( t ) = � { q ( s ) | s ∈ T , sRt } and P ∗ : ( M S ) op → ( M T ) op is a strong fm-function between MV-algebras. We say that G ∗ : M T → M S is the canonical strong fm-function between MV-algebras induced by the frame ( S , T , R ) and the MV-algebra M . Operators on MV-algebras 31 / 42

  76. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Outline Introduction 1 Basic notions and definitions 2 Dyadic numbers and MV-terms 3 Filters, ultrafilters and the term t r 4 Semistates on MV-algebras 5 Functions between MV-algebras and their construction 6 The main theorem and its applications 7 Operators on MV-algebras 32 / 42

  77. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  78. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  79. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  80. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  81. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  82. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Semisimple MV-algebras Recall that semisimple MV-algebras are just subdirect products of the simple 1 MV-algebras, any simple MV-algebra is uniquelly embeddable into the standard 2 MV-algebra on the interval [ 0 , 1 ] of reals, an MV-algebra is semisimple if and only if the intersection of the set of its 3 maximal (prime) filters is equal to the set { 1 } , any complete MV-algebra is semisimple. 4 A semisimple MV-algebra A is embedded into [ 0 , 1 ] T where T is the set of all ultrafilters of A (morphisms from A into the standard MV-algebra ) and π F ( x ) = x ( F ) = x / F ∈ [ 0 , 1 ] for any x ∈ S ⊆ [ 0 , 1 ] T and any F ∈ T ; here π F : [ 0 , 1 ] T → [ 0 , 1 ] is the respective projection onto [ 0 , 1 ] . Operators on MV-algebras 33 / 42

  83. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  84. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  85. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Main theorem Theorem Let G : A 1 → A 2 be an fm-function between semisimple MV-algebras, T ( S ) a set of all MV-morphism from A 1 ( A 2 ) into the standard MV-algebra [ 0 , 1 ] . Further, let ( S , T , ρ G ) be a frame such that the relation ρ G ⊆ S × T is defined by s ρ G t if and only if s ( G ( x )) ≤ t ( x ) for any x ∈ A 1 . Then G is representable via the canonical strong fm-function G ∗ : [ 0 , 1 ] T → [ 0 , 1 ] S between MV-algebras induced by the frame ( S , T , ρ G ) and the standard MV-algebra [ 0 , 1 ] , i.e., the following diagram of fm-functions commutes: G ✲ A 2 A 1 i T i S A 1 A 2 ❄ ❄ ✲ [ 0 , 1 ] S [ 0 , 1 ] T . G ∗ Operators on MV-algebras 34 / 42

  86. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

  87. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

  88. Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications The applications of the main theorem Proposition For any MV-algebra A 1 , any semisimple MV-algebra A 2 with a set S of all MV-morphism from A 2 to [ 0 , 1 ] and any map G : A 1 → A 2 the following conditions are equivalent: (i) G is an fm-function between MV-algebras. (ii) G is a strong fm-function between MV-algebras. Open problem Find MV-algebras A 1 and A 2 with an fm-function G between them such that G is not a strong fm-function. Operators on MV-algebras 35 / 42

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