Variants of finite full transformation semigroups

Dolinka, Igor; East, James

doi:10.1142/s021819671550037x

Cited by 33 publications

(46 citation statements)

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“…The maps in (2.7) were further explored in [9] in the case that S = M(F) was the partial semigroup of (finite dimensional) matrices over a fixed field F; related maps were also studied in [8], in the case that S = T X was the full transformation semigroup on a finite set X. We wish to explore the diagram (2.7) here in more generality.…”

Section: The Basic Commutative Diagramsmentioning

confidence: 99%

Sandwich semigroups in locally small categories I: foundations

et al. 2018

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Fix (not necessarily distinct) objects i and j of a locally small category S, and write S ij for the set of all morphisms i → j. Fix a morphism a ∈ S ji , and define an operation a on S ij by x a y = xay for all x, y ∈ S ij . Then (S ij , a ) is a semigroup, known as a sandwich semigroup, and denoted by S a ij . This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green's relations and stability, focusing on the relationships between these properties on S a ij and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set Reg(S a ij ) of all regular elements of S a ij is a subsemigroup of S a ij . Under this condition, we carefully analyse the structure of the semigroup Reg(S a ij ), relating it via pullback products to certain regular subsemigroups of S ii and S jj , and to a certain regular sandwich monoid defined on a subset of S ji ; among other things, this allows us to also describe the idempotentgenerated subsemigroup E(S a ij ) of S a ij . We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups S a ij , Reg(S a ij ) and E(S a ij ); we give lower bounds for these ranks, and in the case of Reg(S a ij ) and E(S a ij ) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.The concept of stability played a crucial role in [9], and will continue to do so here. Recall that a partial semigroup S is stable if, for all a, x ∈ S,In our studies, we require a more refined notion of stability. We say an element a of a partial semigroupWe say a is stable if it is both R-and L -stable.Recall that a semigroup T is periodic if for each x ∈ T , some power of x is an idempotent: that is, x 2m = x m for some m ≥ 1. It is well known that all finite semigroups are periodic; see for example [25, Proposition 1.2.3]. The proof of the next result is adapted from that of [43, Theorem A.2.4], but is included for convenience. If a ∈ S ji , for some i, j ∈ I, then S ij a is a subsemigroup of S i , and aS ij a subsemigroup of S j . We will have more to say about these subsemigroups in Sections 2 and 3. Lemma 1.3. Let S be a partial semigroup, and fix i, j ∈ I and a ∈ S ji .

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Section: The Basic Commutative Diagramsmentioning

confidence: 99%

Sandwich semigroups in locally small categories I: foundations

et al. 2018

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“…Suppose I is a total ideal of Π(X) and let A be a maximal proper subset of X. Then by the identification in (10) of N * P(X) with Π(X), there is someπ ∈ vI such that A ∈ M H(e; −) where π is the partition induced by e. By Lemma 4.3, this implies that for every proper maximal subset A of X, there existsπ ∈ vI such that A is a cross-section of π.…”

Section: Right Reductive Subsemigroups Of Sing(x)mentioning

confidence: 99%

“…Since every semigroup can be realized as a transformation semigroup, T X and its subsemigroups have been studied extensively [10][11][12][13]. Sing(X) being a regular semigroup with a relatively nice structure makes a good candidate to describe some non-trivial cross-connections because it is not a monoid.…”

Section: Introductionmentioning

confidence: 99%

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Cross-connections of the singular transformation semigroup

Muhammed

Rajan

2018

J. Algebra Appl.

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Abstract. Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup Sing(X) of all non-invertible transformations on a set X. The categories involved are characterized as the powerset category P(X) and the category of partitions Π(X). We describe these categories and show how a permutation on X gives rise to a cross-connection. Further we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to Sing(X). We also describe the right reductive subsemigroups of Sing(X) with the category of principal left ideals isomorphic to P(X). This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of Sing(X).

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“…(ii). It was noted at the beginning of [3,Section 4] that T a X ∼ = T pa X for any permutation p of X, and that there exists such a permutation p for which pa is an idempotent. Thus, without loss of generality, we may assume that a is itself an idempotent (for this part of the proof).…”

Section: Transformation Semigroupsmentioning

confidence: 99%

Transformation Representations of Sandwich Semigroups

East

2018

Experimental Mathematics

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Let a be an element of a semigroup S. The local subsemigroup of S with respect to a is the subsemigroup aSa of S. The variant of S with respect to a is the semigroup with underlying set S and operation a defined by x a y = xay for x, y ∈ S. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semigroups package for GAP.

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Variants of finite full transformation semigroups

Cited by 33 publications

References 63 publications

Sandwich semigroups in locally small categories I: foundations

Sandwich semigroups in locally small categories I: foundations

Cross-connections of the singular transformation semigroup

Transformation Representations of Sandwich Semigroups

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