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Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Kittisak Tinpun Institute of Mathematics, Am Neuen Palais 10, University of Potsdam, 14469 Potsdam, Germany 20-22 June 2014 1 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Outline Outline 1 Background and Motivation 2 Introduction 3 Preliminaries and Notations 4 Main Results 2 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Background and Motivation Background and Motivation WHY ??? 3 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Full transformation Let X be a non-empty set. Then a mapping α from set X to set X is called full transformation . • The image of α : imα = { xα : x ∈ X } . Then T ( X ) is the set of all full transformation semigroup under composition. 4 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Full transformation Let X be a non-empty set. Then a mapping α from set X to set X is called full transformation . • The image of α : imα = { xα : x ∈ X } . Then T ( X ) is the set of all full transformation semigroup under composition. 5 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Rank of semigroup S rankS := min {| A | : A ⊆ S, < A > = S } . Example 1 The ranks of well known semigroups have been calculated by J.M.Howie (1995): • A finite full transformation semigroup has rank 3; • A finite partial transformation semigroup has rank 4. 6 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Rank of semigroup S rankS := min {| A | : A ⊆ S, < A > = S } . Example 1 The ranks of well known semigroups have been calculated by J.M.Howie (1995): • A finite full transformation semigroup has rank 3; • A finite partial transformation semigroup has rank 4. 7 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Relative rank of S modulo A In the case of X is infinite, we have T ( X ) is uncountable. Then Howie and Ruskuc (1998) defined the relative rank of S modulo A ⊆ S by rank ( S : A ) := min {| B | : B ⊆ S, < A ∪ B > = S } . From the definition, we have: • rank ( S : ∅ ) = rankS ; • rank ( S : S ) = 0; • rank ( S : A ) = 0 if and only if A is a generating set for S . 8 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Relative rank of S modulo A In the case of X is infinite, we have T ( X ) is uncountable. Then Howie and Ruskuc (1998) defined the relative rank of S modulo A ⊆ S by rank ( S : A ) := min {| B | : B ⊆ S, < A ∪ B > = S } . From the definition, we have: • rank ( S : ∅ ) = rankS ; • rank ( S : S ) = 0; • rank ( S : A ) = 0 if and only if A is a generating set for S . 9 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Transformation semigroup T ( X, Y ) Let X be an infinite set and let Y be a non-empty subset of X . Then T ( X, Y ) was introduced by Symons (1975) to be the set of all full transformation from set X to set Y that means T ( X, Y ) := { α ∈ T ( X ) : Xα ⊆ Y } . Clearly, T ( X, Y ) is a subsemigroup of T ( X ) and if X = Y then T ( X, Y ) = T ( X ). 10 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Introduction Transformation semigroup T ( X, Y ) Let X be an infinite set and let Y be a non-empty subset of X . Then T ( X, Y ) was introduced by Symons (1975) to be the set of all full transformation from set X to set Y that means T ( X, Y ) := { α ∈ T ( X ) : Xα ⊆ Y } . Clearly, T ( X, Y ) is a subsemigroup of T ( X ) and if X = Y then T ( X, Y ) = T ( X ). 11 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations rankα and d ( α ) in T ( X, Y ) Let α ∈ T ( X, Y ). Then we can define various parameters associated to a mapping in T ( X, Y ). Definition 1 Let α ∈ T ( X, Y ). Then rankα is defined to be the cardinality of image of α , i.e. rankα := | imα | . Definition 2 Let α ∈ T ( X, Y ). Then defect of α is defined by d ( α ) := | Y \ imα | . 12 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations rankα and d ( α ) in T ( X, Y ) Let α ∈ T ( X, Y ). Then we can define various parameters associated to a mapping in T ( X, Y ). Definition 1 Let α ∈ T ( X, Y ). Then rankα is defined to be the cardinality of image of α , i.e. rankα := | imα | . Definition 2 Let α ∈ T ( X, Y ). Then defect of α is defined by d ( α ) := | Y \ imα | . 13 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations rankα and d ( α ) in T ( X, Y ) Let α ∈ T ( X, Y ). Then we can define various parameters associated to a mapping in T ( X, Y ). Definition 1 Let α ∈ T ( X, Y ). Then rankα is defined to be the cardinality of image of α , i.e. rankα := | imα | . Definition 2 Let α ∈ T ( X, Y ). Then defect of α is defined by d ( α ) := | Y \ imα | . 14 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations c ( α ) in T ( X, Y ) Consider the kernel relation of α ; ker α := { ( x, y ) ∈ X × X : xα = yα } . Clearly, ker α is an equivalence relation on X . Definition 3 A transversal of α ∈ T ( X, Y ) is any set T α ⊆ X such that T α α = imα and α | T α is 1-1 (i.e. a transversal of the equivalence classes of ker α ). Definition 4 Let α ∈ T ( X, Y ) and T α be a transversal of ker α . Then the collapse of α is defined by c ( α ) := | X \ T α | 15 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations c ( α ) in T ( X, Y ) Consider the kernel relation of α ; ker α := { ( x, y ) ∈ X × X : xα = yα } . Clearly, ker α is an equivalence relation on X . Definition 3 A transversal of α ∈ T ( X, Y ) is any set T α ⊆ X such that T α α = imα and α | T α is 1-1 (i.e. a transversal of the equivalence classes of ker α ). Definition 4 Let α ∈ T ( X, Y ) and T α be a transversal of ker α . Then the collapse of α is defined by c ( α ) := | X \ T α | 16 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations c ( α ) in T ( X, Y ) Consider the kernel relation of α ; ker α := { ( x, y ) ∈ X × X : xα = yα } . Clearly, ker α is an equivalence relation on X . Definition 3 A transversal of α ∈ T ( X, Y ) is any set T α ⊆ X such that T α α = imα and α | T α is 1-1 (i.e. a transversal of the equivalence classes of ker α ). Definition 4 Let α ∈ T ( X, Y ) and T α be a transversal of ker α . Then the collapse of α is defined by c ( α ) := | X \ T α | 17 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations k ( α ) in T ( X, Y ) Definition 5 Let α ∈ T ( X, Y ). Then the infinite contraction index of mapping α is defined as the number of classes of ker α of size | X | . If X = Y , then the definition of rank of α , the defect of α , the collapse of α , and the infinite contraction index of α coinside with the usual ones. Lemma 1 Let Y be a non-empty subset of X and α, β ∈ T ( Y ) = T ( Y, Y ). Then we have (i) d ( αβ ) ≤ d ( α ) + d ( β ); (ii) If | Y | is a regular cardinal then k ( αβ ) ≤ k ( α ) + k ( β ). 18 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations k ( α ) in T ( X, Y ) Definition 5 Let α ∈ T ( X, Y ). Then the infinite contraction index of mapping α is defined as the number of classes of ker α of size | X | . If X = Y , then the definition of rank of α , the defect of α , the collapse of α , and the infinite contraction index of α coinside with the usual ones. Lemma 1 Let Y be a non-empty subset of X and α, β ∈ T ( Y ) = T ( Y, Y ). Then we have (i) d ( αβ ) ≤ d ( α ) + d ( β ); (ii) If | Y | is a regular cardinal then k ( αβ ) ≤ k ( α ) + k ( β ). 19 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Preliminaries and Notations k ( α ) in T ( X, Y ) Definition 5 Let α ∈ T ( X, Y ). Then the infinite contraction index of mapping α is defined as the number of classes of ker α of size | X | . If X = Y , then the definition of rank of α , the defect of α , the collapse of α , and the infinite contraction index of α coinside with the usual ones. Lemma 1 Let Y be a non-empty subset of X and α, β ∈ T ( Y ) = T ( Y, Y ). Then we have (i) d ( αβ ) ≤ d ( α ) + d ( β ); (ii) If | Y | is a regular cardinal then k ( αβ ) ≤ k ( α ) + k ( β ). 20 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Main Results rank ( T ( X, Y ) : J 1 ) =??? Let J 1 := { α ∈ T ( X, Y ) : rankα | Y = | Y |} . Proposition 1 Let X be an infinite set with regular cardinality and let Y be a non-empty subset of X . Then < J 1 > = T ( X, Y ) and rank ( T ( X, Y ) : J 1 ) = 0 21 / 37

Relative Ranks of Infinite Full Transformation Semigroups T ( X, Y ) Main Results rank ( T ( X, Y ) : J 1 ) =??? Let J 1 := { α ∈ T ( X, Y ) : rankα | Y = | Y |} . Proposition 1 Let X be an infinite set with regular cardinality and let Y be a non-empty subset of X . Then < J 1 > = T ( X, Y ) and rank ( T ( X, Y ) : J 1 ) = 0 22 / 37