Semigroups of order-preserving transformations Young researchers in - PowerPoint PPT Presentation

Semigroups of order-preserving transformations Young researchers in mathematics 3 rd August 2017 Wilf Wilson University of St Andrews Wilf Wilson Semigroups of order-preserving transformations Page 0 of 412

What are semigroups and monoids? Informally, a binary operation is a way of combining two elements to give a third, such as + − × ÷ in R . A binary operation is associative if x · ( y · z ) = ( x · y ) · z for all possible x , y , z . Definition (Semigroup) A semigroup is a set with an associative binary operation. Definition (Monoid) A monoid is a semigroup with an identity element, 1. i.e. 1 · x = x · 1 = x , for all elements x . Wilf Wilson Semigroups of order-preserving transformations Page 1 of 412

What are semigroups and monoids? Informally, a binary operation is a way of combining two elements to give a third, such as + − × ÷ in R . A binary operation is associative if x · ( y · z ) = ( x · y ) · z for all possible x , y , z . Definition (Semigroup) A semigroup is a set with an associative binary operation. Definition (Monoid) A monoid is a semigroup with an identity element, 1. i.e. 1 · x = x · 1 = x , for all elements x . Wilf Wilson Semigroups of order-preserving transformations Page 1 of 412

What are semigroups and monoids? Informally, a binary operation is a way of combining two elements to give a third, such as + − × ÷ in R . A binary operation is associative if x · ( y · z ) = ( x · y ) · z for all possible x , y , z . Definition (Semigroup) A semigroup is a set with an associative binary operation. Definition (Monoid) A monoid is a semigroup with an identity element, 1. i.e. 1 · x = x · 1 = x , for all elements x . Wilf Wilson Semigroups of order-preserving transformations Page 1 of 412

Semigroups and monoids can be formed of. . . ◮ Square matrices over a ring R , with multiplication. ◮ Binary relations, with composition. ◮ Subsets of N , with addition. ◮ Endomorphisms of a structure, with composition. ◮ If A = { a 1 , . . . , a m } , then: � A ∗ = { all strings over A } is the free monoid over A . � A + = A ∗ \ { empty string } is the free semigroup over A . (With the operation of concatenation.) Wilf Wilson Semigroups of order-preserving transformations Page 2 of 412

Semigroups and monoids can be formed of. . . ◮ Square matrices over a ring R , with multiplication. ◮ Binary relations, with composition. ◮ Subsets of N , with addition. ◮ Endomorphisms of a structure, with composition. ◮ If A = { a 1 , . . . , a m } , then: � A ∗ = { all strings over A } is the free monoid over A . � A + = A ∗ \ { empty string } is the free semigroup over A . (With the operation of concatenation.) Wilf Wilson Semigroups of order-preserving transformations Page 2 of 412

How to specify a semigroup ◮ Multiplication table: · a b c a a a a b a b b c a c c ◮ Generating set: S = � 13, 8 � � N ◮ Semigroup presentation: � x 1 , x 2 | x 4 1 = x 2 x 1 � ◮ . . . and all sorts of algebraic ways. Wilf Wilson Semigroups of order-preserving transformations Page 3 of 412

What do I do? ◮ Computational methods for finite semigroups. Given a generating set, you might want to: ◮ Compute the size of the semigroup. ◮ Test membership in the semigroup. ◮ Describe maximal subgroups/subsemigroups. ◮ Calculate ideals. ◮ Compute congruences. ◮ Find the idempotents. ◮ Work out whether there is an identity/zero. Algorithms typically have bad worst-case complexity. Wilf Wilson Semigroups of order-preserving transformations Page 4 of 412

Transformations Definition (Transformation) A transformation is a map from { 1, . . . , n } to { 1, . . . , n } . � 1 � 2 · · · n We can write f = . 1 f 2 f · · · nf Definition (Partial transformation) A partial transformation is a partial map on { 1, . . . , n } . � 1 � 2 3 4 5 We can write f = , for example. − 4 1 − 4 Definition (Partial permutation) A partial permutation is an injective partial transformation. Wilf Wilson Semigroups of order-preserving transformations Page 5 of 412

Transformations Definition (Transformation) A transformation is a map from { 1, . . . , n } to { 1, . . . , n } . � 1 � 2 · · · n We can write f = . 1 f 2 f · · · nf Definition (Partial transformation) A partial transformation is a partial map on { 1, . . . , n } . � 1 � 2 3 4 5 We can write f = , for example. − 4 1 − 4 Definition (Partial permutation) A partial permutation is an injective partial transformation. Wilf Wilson Semigroups of order-preserving transformations Page 5 of 412

Transformations Definition (Transformation) A transformation is a map from { 1, . . . , n } to { 1, . . . , n } . � 1 � 2 · · · n We can write f = . 1 f 2 f · · · nf Definition (Partial transformation) A partial transformation is a partial map on { 1, . . . , n } . � 1 � 2 3 4 5 We can write f = , for example. − 4 1 − 4 Definition (Partial permutation) A partial permutation is an injective partial transformation. Wilf Wilson Semigroups of order-preserving transformations Page 5 of 412

Image, domain, kernel Let α be a partial transformation on the set { 1, . . . , n } . ◮ im ( α ) = � � i α : i ∈ { 1, . . . , n } . ◮ dom ( α ) = � � i ∈ { 1, . . . , n } : i α is defined . ◮ ker ( α ) is the partition of dom ( α ) into parts with equal image. � 1 � 2 3 4 5 Example: if α = , then − 4 1 − 4 ◮ im ( α ) = { 1, 4 } , ◮ dom ( α ) = { 2, 3, 5 } , ◮ ker ( α ) = � � { 2, 5 } , { 3 } . These attributes are often useful for computations. Wilf Wilson Semigroups of order-preserving transformations Page 6 of 412

Image, domain, kernel Let α be a partial transformation on the set { 1, . . . , n } . ◮ im ( α ) = � � i α : i ∈ { 1, . . . , n } . ◮ dom ( α ) = � � i ∈ { 1, . . . , n } : i α is defined . ◮ ker ( α ) is the partition of dom ( α ) into parts with equal image. � 1 � 2 3 4 5 Example: if α = , then − 4 1 − 4 ◮ im ( α ) = { 1, 4 } , ◮ dom ( α ) = { 2, 3, 5 } , ◮ ker ( α ) = � � { 2, 5 } , { 3 } . These attributes are often useful for computations. Wilf Wilson Semigroups of order-preserving transformations Page 6 of 412

Image, domain, kernel Let α be a partial transformation on the set { 1, . . . , n } . ◮ im ( α ) = � � i α : i ∈ { 1, . . . , n } . ◮ dom ( α ) = � � i ∈ { 1, . . . , n } : i α is defined . ◮ ker ( α ) is the partition of dom ( α ) into parts with equal image. � 1 � 2 3 4 5 Example: if α = , then − 4 1 − 4 ◮ im ( α ) = { 1, 4 } , ◮ dom ( α ) = { 2, 3, 5 } , ◮ ker ( α ) = � � { 2, 5 } , { 3 } . These attributes are often useful for computations. Wilf Wilson Semigroups of order-preserving transformations Page 6 of 412

Transformation semigroups Composition of functions is associative! A transformation semigroup is any semigroup of transformations, with composition of functions. ◮ PT n , the partial transformation semigroup of degree n , is the semigroup consisting of all partial transformations on { 1, . . . , n } . ◮ T n , the full transformation semigroup of degree n , is the semigroup consisting of all transformations on { 1, . . . , n } . There’s an analogue of Cayley’s theorem here. Wilf Wilson Semigroups of order-preserving transformations Page 7 of 412

Transformation semigroups Composition of functions is associative! A transformation semigroup is any semigroup of transformations, with composition of functions. ◮ PT n , the partial transformation semigroup of degree n , is the semigroup consisting of all partial transformations on { 1, . . . , n } . ◮ T n , the full transformation semigroup of degree n , is the semigroup consisting of all transformations on { 1, . . . , n } . There’s an analogue of Cayley’s theorem here. Wilf Wilson Semigroups of order-preserving transformations Page 7 of 412

Inverse semigroups Definition (Inverse semigroup) A semigroup S is inverse if all for x ∈ S , there is a unique x ′ ∈ S such that x = xx ′ x and x ′ = x ′ xx ′ . ◮ I n , the symmetric inverse monoid of degree n , is the semigroup consisting of all partial permutations on { 1, . . . , n } . There’s another analogue of Cayley’s theorem . Wilf Wilson Semigroups of order-preserving transformations Page 8 of 412

Inverse semigroups Definition (Inverse semigroup) A semigroup S is inverse if all for x ∈ S , there is a unique x ′ ∈ S such that x = xx ′ x and x ′ = x ′ xx ′ . ◮ I n , the symmetric inverse monoid of degree n , is the semigroup consisting of all partial permutations on { 1, . . . , n } . There’s another analogue of Cayley’s theorem . Wilf Wilson Semigroups of order-preserving transformations Page 8 of 412

The sizes of these semigroups ◮ All permutations: |S n | = n ! n � n � n k = ( n + 1 ) n . � ◮ Partial transformations: |PT n | = k k = 0 ◮ All transformations: |T n | = n n . n � 2 � n � ◮ All partial permutations: |I n | = k ! k k = 0 Wilf Wilson Semigroups of order-preserving transformations Page 9 of 412

The sizes of these semigroups ◮ All permutations: |S n | = n ! n � n � n k = ( n + 1 ) n . � ◮ Partial transformations: |PT n | = k k = 0 ◮ All transformations: |T n | = n n . n � 2 � n � ◮ All partial permutations: |I n | = k ! k k = 0 Wilf Wilson Semigroups of order-preserving transformations Page 9 of 412