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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

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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

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Background and Motivation

Background and Motivation

WHY ???

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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).

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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

• f 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|.

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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

• f 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|.

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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

• f 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|.

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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 Tα ⊆ X such that Tαα = imα and α|Tα is 1-1 (i.e. a transversal of the equivalence classes of ker α). Definition 4 Let α ∈ T(X, Y ) and Tα be a transversal of ker α. Then the collapse of α is defined by c(α) := |X \ Tα|

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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 Tα ⊆ X such that Tαα = imα and α|Tα is 1-1 (i.e. a transversal of the equivalence classes of ker α). Definition 4 Let α ∈ T(X, Y ) and Tα be a transversal of ker α. Then the collapse of α is defined by c(α) := |X \ Tα|

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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 Tα ⊆ X such that Tαα = imα and α|Tα is 1-1 (i.e. a transversal of the equivalence classes of ker α). Definition 4 Let α ∈ T(X, Y ) and Tα be a transversal of ker α. Then the collapse of α is defined by c(α) := |X \ Tα|

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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).

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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).

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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).

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : J1) =???

Let J1 := {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 < J1 >= T(X, Y ) and rank(T(X, Y ) : J1) = 0

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : J1) =???

Let J1 := {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 < J1 >= T(X, Y ) and rank(T(X, Y ) : J1) = 0

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

S(X, Y ) := {α ∈ T(X, Y ) : α|Y ∈ S(Y )} ; where S(Y ) is a symmetric group on set Y . Proposition 2 Let X be an infinite set and let Y be infinite subset of X and let ¯ µ ∈ T(X, Y ) be arbitrary mapping. Then < S(X, Y ), ¯ µ >= T(X, Y ), i.e. rank(T(X, Y ) : S(X, Y )) > 1.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

S(X, Y ) := {α ∈ T(X, Y ) : α|Y ∈ S(Y )} ; where S(Y ) is a symmetric group on set Y . Proposition 2 Let X be an infinite set and let Y be infinite subset of X and let ¯ µ ∈ T(X, Y ) be arbitrary mapping. Then < S(X, Y ), ¯ µ >= T(X, Y ), i.e. rank(T(X, Y ) : S(X, Y )) > 1.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

Theorem 1 Let X be an infinite set and let Y be an infinite subset of X, let ¯ µ ∈ T(X, Y ) with ¯ µ|Y be an injection of defect |Y | and let ¯ ν ∈ T(X, Y ) with ¯ ν|Y be a surjection of infinite contraction index |Y |. Then < S(X, Y ), ¯ µ, ¯ ν >= T(X, Y ), i.e. rank(T(X, Y ) : S(X, Y )) = 2. Theorem 2 Let X be an infinite set and let Y be an infinite subset of X with regular cardinality, and let ¯ µ, ¯ ν ∈ T(X, Y ). Then < S(X, Y ), ¯ µ, ¯ ν >= T(X, Y ) if and only if ¯ µ|Y is injective of defect |Y | and ¯ ν|Y is surjective of infinite contraction index |Y |.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

Theorem 1 Let X be an infinite set and let Y be an infinite subset of X, let ¯ µ ∈ T(X, Y ) with ¯ µ|Y be an injection of defect |Y | and let ¯ ν ∈ T(X, Y ) with ¯ ν|Y be a surjection of infinite contraction index |Y |. Then < S(X, Y ), ¯ µ, ¯ ν >= T(X, Y ), i.e. rank(T(X, Y ) : S(X, Y )) = 2. Theorem 2 Let X be an infinite set and let Y be an infinite subset of X with regular cardinality, and let ¯ µ, ¯ ν ∈ T(X, Y ). Then < S(X, Y ), ¯ µ, ¯ ν >= T(X, Y ) if and only if ¯ µ|Y is injective of defect |Y | and ¯ ν|Y is surjective of infinite contraction index |Y |.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

Moreover, we can show that the relative rank of T(X, Y ) modulo S(X, Y ) is infinite in the case 2 ≤ |Y | < ℵ0. Proposition 3 Let X be an infinite set and let Y be a finite subset of X with at least two elements. Then rank(T(X, Y ) : S(X, Y )) is infinite.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank(T(X, Y ) : S(X, Y )) =???

Moreover, we can show that the relative rank of T(X, Y ) modulo S(X, Y ) is infinite in the case 2 ≤ |Y | < ℵ0. Proposition 3 Let X be an infinite set and let Y be a finite subset of X with at least two elements. Then rank(T(X, Y ) : S(X, Y )) is infinite.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : E(X, Y )) =???

Example 2 Let X = N and Y = {1, 2}. Then the relative rank of T(X, Y ) modulo E(X, Y ) is infinite. Proposition 4 Let X be an infinite set and Y be a finite subset of X. Then rank(T(X, Y ) : E(X, Y )) is infinite.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : E(X, Y )) =???

Example 2 Let X = N and Y = {1, 2}. Then the relative rank of T(X, Y ) modulo E(X, Y ) is infinite. Proposition 4 Let X be an infinite set and Y be a finite subset of X. Then rank(T(X, Y ) : E(X, Y )) is infinite.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : E(X, Y )) =???

Example 2 Let X = N and Y = {1, 2}. Then the relative rank of T(X, Y ) modulo E(X, Y ) is infinite. Proposition 4 Let X be an infinite set and Y be a finite subset of X. Then rank(T(X, Y ) : E(X, Y )) is infinite.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

rank (T(X, Y ) : E(X, Y )) =???

Theorem 3 Let X be an infinite set and let Y be infinite subset of X, let ¯ µ ∈ T(X, Y ) with ¯ µ|Y is an injection with defect |Y | and let ¯ ν be any mapping extending (¯ µ|Y )−1. Then < E(X, Y ), ¯ µ, ¯ ν >= T(X, Y ) and rank(T(X, Y ) : E(X, Y )) = 2.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

• J. M. Howie (1966) characterized the transformations in

T(X) generated by the idempotents in T(X). Theorem 4 (Howie, 1966) Let T(X) be the full transformation semigroup

• n an infinite set X. The subsemigroup E(X) of T(X)

generated by the idempotents of non-zero defect consists of all elements finite shift and finite non-zero defect together with those element α of infinite shift |s(α)| = |d(α)| = |c(α)|. Proposition 5 Let δ ∈ T(X, Y ) with c(δ|Y ) = |Y | and d(δ|Y ) = |Y | such that there exists a transversal Tδ of ker δ in Y . Then δ ∈< E(X, Y ) >.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

• J. M. Howie (1966) characterized the transformations in

T(X) generated by the idempotents in T(X). Theorem 4 (Howie, 1966) Let T(X) be the full transformation semigroup

• n an infinite set X. The subsemigroup E(X) of T(X)

generated by the idempotents of non-zero defect consists of all elements finite shift and finite non-zero defect together with those element α of infinite shift |s(α)| = |d(α)| = |c(α)|. Proposition 5 Let δ ∈ T(X, Y ) with c(δ|Y ) = |Y | and d(δ|Y ) = |Y | such that there exists a transversal Tδ of ker δ in Y . Then δ ∈< E(X, Y ) >.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

• J. M. Howie (1966) characterized the transformations in

T(X) generated by the idempotents in T(X). Theorem 4 (Howie, 1966) Let T(X) be the full transformation semigroup

• n an infinite set X. The subsemigroup E(X) of T(X)

generated by the idempotents of non-zero defect consists of all elements finite shift and finite non-zero defect together with those element α of infinite shift |s(α)| = |d(α)| = |c(α)|. Proposition 5 Let δ ∈ T(X, Y ) with c(δ|Y ) = |Y | and d(δ|Y ) = |Y | such that there exists a transversal Tδ of ker δ in Y . Then δ ∈< E(X, Y ) >.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

Main Theorem

Theorem 5 Let X be an infinite set and let Y be an infinite subset of X. Then we have < E(X, Y ), ¯ µ, ¯ ν >= T(X, Y ) if and only if one of the mappings ¯ µ and ¯ ν restricted to Y is an injection with defect |Y | and the other is a surjection with collapse |Y |.

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Relative Ranks of Infinite Full Transformation Semigroups T (X, Y ) Main Results

THANK YOU FOR YOUR ATTENTION.

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