Regular Subsemigroups Research Articles

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(Tr), and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.

Regular semigroups weakly generated by one element

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup F_{1} weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F_{1}. We define F_{1} using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of F_{1}. In particular, we show that the ‘free regular semigroup {textrm{FI}}_2 weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of F_{1} weakly generated by {xx',x'x}.

Regular semigroups weakly generated by idempotents

A regular semigroup is weakly generated by a set [Formula: see text] if it has no proper regular subsemigroup containing [Formula: see text]. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup [Formula: see text] weakly generated by [Formula: see text] idempotents such that all other regular semigroups weakly generated by [Formula: see text] idempotents are homomorphic images of [Formula: see text]. The semigroup [Formula: see text] is defined by a presentation [Formula: see text] and its structure is studied. Although each of the sets [Formula: see text], [Formula: see text], and [Formula: see text] is infinite for [Formula: see text], we show that the word problem is decidable as each congruence class has a “canonical form”. If [Formula: see text] denotes [Formula: see text] for [Formula: see text], we prove also that [Formula: see text] contains copies of all [Formula: see text] as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide [Formula: see text]; and (ii) all finite semigroups divide [Formula: see text].

Regularity and maximal subsemigroups in semigroups of full transformations with fixed permuting sets

Let [Formula: see text] be a nonempty set and [Formula: see text] denote the full transformation semigroup on [Formula: see text]. For a fixed nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] where [Formula: see text] is a permutation group on [Formula: see text]. Then [Formula: see text] is a regular submonoid of [Formula: see text]. In this paper, we describe all intra-regular and unit regular elements of [Formula: see text] and give necessary and sufficient conditions for [Formula: see text] to be intra-regular and unit regular. We also count the number of these elements when [Formula: see text] is a finite set. Moreover, we classify the maximal subsemigroups of [Formula: see text] and prove that these maximal subsemigroup coincide with the maximal regular subsemigroups of [Formula: see text] when [Formula: see text] is a finite set.

Categories and semigroups

The aim of this paper is to consider the correspondence between the classification of morphisms in categories and the classes of semigroups with idempotents, in particular, we establish a mutual corresponding theorem of three classes of categories and the classes of left (right) abundant, two-sided abundant semigroups and regular semigroups. We first apply the nine axioms (P1)–(P9) which characterize cancellation and split properties of morphisms in a category to classify categories into three subclasses: idempotent ample ([Formula: see text]-, for short) categories, balanced categories and normal categories. The intrinsic relationship between these categories and their cone semigroups is investigated. It is proved that the cone semigroup of an [Formula: see text]-category is left abundant and vice versa, the category of principal left ∗-ideals of a left abundant (not necessarily right abundant) semigroup is an [Formula: see text]-category. Similar results for balanced categories and abundant semigroups are reproved and strengthened. As a consequence, we employ categorical terms to characterize those abundant semigroups in which the regular elements form a regular subsemigroup. Therefore, the significant theory of normal categories and regular semigroups devoted by K. S. S. Nambooripad is generalized to arbitrary abundant semigroups and even to left and right abundant semigroups, respectively.

Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range

UDC 512.64 Let be a vector space and let denote the semigroup (under composition) of all linear transformations from into . For a fixed subspace of , let be the semigroup consisting of all linear transformations from into . It is known that is the largest regular subsemigroup of . In this paper, we prove that any regular semigroup can be embedded in with and , and determine all the maximal subsemigroups of when is a finite dimensional subspace of over a finite field.

Isbell’s Zigzag theorem for permutative orthodox semigroups and clifford semigroups

In this paper, we prove that the dominion of any full orthodox subsemigroup of a medial orthodox semigroup is described by the Isbell zigzag theorem in the category of medial orthodox semigroups. As a consequence, the dominions of any full completely regular subsemigroup of a medial completely regular semigroup as well as that of any full Clifford subsemigroup of a medial Clifford semigroup are also described by the Isbell zigzag theorem in the categories of all medial completely regular semigroups and of all medial Clifford semigroups, respectively.

Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

Some Properties of the semigroup PGY (X): Green’s relations, ideals, isomorphism theorems and ranks

Let $T(X)$ be the full transformation semigroup on the set $X$. For a fixed nonempty subset $Y$ of $X$, let \begin{equation*} PG_Y(X) = \{\alpha\in T(X) : \alpha _Y\in G(Y)\} \end{equation*} where $G(Y)$ is the permutation group on $Y$. It is known that $PG_Y(X)$ is a regular subsemigroup of $T(X)$. In this paper, we give a simpler description of Green's relations and characterize the ideals of $PG_Y(X)$. Moreover, we prove some isomorphism theorems for $PG_Y(X)$. For finite sets, we investigate the cardinalities of $PG_Y(X)$ and of its subsets of idempotents, and we also calculate their ranks.

On certain semigroups of contraction mappings of a finite chain

Let[n] ={1,2, . . . , n} be a finite chain and let Pn (resp.,Tn) be the semigroup of partial transformations on[n] (resp., full transformations on[n]). Let CPn={α∈ Pn: (for allx, y ∈ Dom α)|xα−yα|⩽|x−y|}(resp., CTn={α∈ Tn: (for allx, y∈[n])|xα−yα|⩽|x−y|}) be the subsemigroup of partial contractionmappings on[n](resp., subsemigroup of full contraction mappingson[n]). We characterize all the starred Green’s relations on C Pn and it subsemigroup of order preserving and/or order reversingand subsemigroup of order preserving partial contractions on[n], respectively. We show that the semigroups CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant forn⩾4. We further show that the set ofregular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on[n], each formsa regular subsemigroup and an orthodox semigroup, respectively.

Cross-connection structure of concordant semigroups

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

Regularity in the Semigroup of Transformations Preserving a Zig-Zag Order

It is well known that the semigroup T(X) of all transformations on a set X is regular, but its subsemigroups do not need to be. Consider an ordered set $$(X;\le )$$ whose order forms a path with alternating orientation. The semigroup $$\mathrm{OT}(X)$$ of all order-preserving transformations on $$(X;\le )$$ is a subsemigroup of T(X). Some characterizations of regular semigroups $$\mathrm{OT}(X)$$ were given. For a non-empty subset S of X, we define the set $$\begin{aligned} \mathrm{OT}_S(X)=\{\alpha \in \mathrm{OT}(X)\mid {{\,\mathrm{ran}\,}}\alpha =S \}. \end{aligned}$$In this paper, we show that in general, $$\mathrm{OT}_S(X)$$ is not a subsemigroup of T(X). A necessary and sufficient condition for $$\mathrm{OT}_S(X)$$ to be a regular subsemigroup of T(X) is given. Some regular subsemigroups of T(X) that are contained in $$\mathrm{OT}_S(X)$$ are studied. We investigate the left (right) regularity of $$\mathrm{OT}(X)$$. We obtain that left regular and right regular elements in $$\mathrm{OT}(X)$$ are identical. A characterization of left (right) elements in $$\mathrm{OT}(X)$$ is given. We classify all left (right) regular semigroups $$\mathrm{OT}(X)$$.

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write $$\mathcal P\mathcal T_{XY}$$ for the set of all partial functions $$X\rightarrow Y$$ . Fix a partial function $${a:Y\rightarrow X}$$ , and define the operation $$\star _a$$ on $$\mathcal P\mathcal T_{XY}$$ by $$f\star _ag=fag$$ for $$f,g\in \mathcal P\mathcal T_{XY}$$ . The sandwich semigroup $$(\mathcal P\mathcal T_{XY},\star _a)$$ is denoted $$\mathcal P\mathcal T_{XY}^a$$ . We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $$\mathcal P\mathcal T_{XY}^a$$ , as well as its regular and idempotent-generated subsemigroups, $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . After describing regularity, stability and Green’s relations and preorders, we exhibit $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups $$\mathcal P\mathcal T_X$$ and $$\mathcal P\mathcal T_Y$$ , and as a kind of “inflation” of $$\mathcal P\mathcal T_A$$ , where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of $$\mathcal P\mathcal T_{XY}^a$$ , $${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$$ and $$\mathbb E(\mathcal P\mathcal T_{XY}^a)$$ . The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write $$S_{ij}$$ for the set of all morphisms $$i\rightarrow j$$ . Fix a morphism $$a\in S_{ji}$$ , and define an operation $$\star _a$$ on $$S_{ij}$$ by $$x\star _ay=xay$$ for all $$x,y\in S_{ij}$$ . Then $$(S_{ij},\star _a)$$ is a semigroup, known as a sandwich semigroup, and denoted by $$S_{ij}^a$$ . 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 $$S_{ij}^a$$ and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set $${\text {Reg}}(S_{ij}^a)$$ of all regular elements of $$S_{ij}^a$$ is a subsemigroup of $$S_{ij}^a$$ . Under this condition, we carefully analyse the structure of the semigroup $${\text {Reg}}(S_{ij}^a)$$ , 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 idempotent-generated subsemigroup $$\mathbb E(S_{ij}^a)$$ of $$S_{ij}^a$$ . We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups $$S_{ij}^a$$ , $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ ; we give lower bounds for these ranks, and in the case of $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ 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 Regular Part of a Semigroup of Full Transformations with Restricted Range: Maximal Inverse Subsemigroups and Maximal Regular Subsemigroups of Its Ideals

Let TX be the full transformation semigroup on a set X. For a fixed nonempty subset Y of a set X, let TX,Y be the semigroup consisting of all full transformations from X into Y. In a paper published in 2008, Sanwong and Sommanee proved that the set FX,Y=α∈TX,Y:Xα=Yα is the largest regular subsemigroup of TX,Y. In this paper, we describe the maximal inverse subsemigroups of FX,Y and completely determine all the maximal regular subsemigroups of its ideals.

Maximal subsemigroups and finiteness conditions on transformation semigroups with fixed sets

Let $Y$ be a fixed subset of a nonempty set $X$ and let $Fix(X,Y)$ be the set of all self maps on $X$ which fix all elements in $Y$. Then under the composition of maps, $Fix(X,Y)$ is a regular monoid. In this paper, we prove that there are only three types of maximal subsemigroups of $Fix\left(X,Y\right)$ and these maximal subsemigroups coincide with the maximal regular subsemigroups when $X\setminus Y$ is a finite set with $|X\setminus Y|\geq 2$. We also give necessary and sufficient conditions for $Fix(X,Y)$ to be factorizable, unit-regular, and directly finite.

Maximal Regular Subsemibands of Finite Orientation-Preserving Partial Transformation Semigroup

Let \(\mathcal {POP}_n\) be the semigroup of all orientation-preserving partial transformations on the finite chain \(X_n=\{1<2<\cdots <n\}\). For \(1\le r\le n-1\), set \(\mathcal {POP}(n, r)=\{\alpha \in \mathcal {POP}_n {:} \, |\mathop {\mathrm {im}}\nolimits (\alpha )|\le r\}\). In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,r)\). First, we completely describe the maximal regular subsemigroups of the semigroup \(\mathcal {POP}(n,r)\), for \(2\le r\le n-1\). Secondly, we show that, for \(2\le r \le n-2\), any maximal regular subsemigroup of the semigroup \(\mathcal {POP}(n,r)\) is a semiband and prove that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,r)\) coincide, for \(2\le r\le n-2\). Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup \(\mathcal {POP}(n,n-1)\).

Maximal Regular Subsemibands of the Finite Order-Preserving Partial Transformation Semigroups $$\mathcal {PO}(n,r)$$ PO ( n , r )

Let $$\mathcal {PO}_n$$ be the semigroup of all order-preserving partial transformations on the finite set $$X_n=\{1, 2,\ldots , n\}$$ . For $$1\le r\le n-1$$ , set $$\mathcal {PO}(n, r)=\{\alpha \in \mathcal {PO}_n: |\mathop {\text{ im }}\nolimits (\alpha )|\le r\}$$ . In this paper, we investigate the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ . First, we completely describe the maximal regular subsemigroups of the semigroup $$\mathcal {PO}(n,r)$$ , for $$1\le r\le n-1$$ . Secondly, we show that, for $$2\le r \le n-2$$ , any maximal regular subsemigroup of the semigroup $$\mathcal {PO}(n,r)$$ is a semiband and obtain that the maximal regular subsemigroups and the maximal regular subsemibands of the semigroup $$\mathcal {PO}(n,r)$$ coincide, for $$2\le r\le n-2$$ . Finally, we obtain the complete classification of maximal regular subsemibands of the semigroup $$\mathcal {PO}_n$$ .

A Munn type representation of abundant semigroups with a multiplicative ample transversal

The celebrated construction by Munn of a fundamental inverse semigroup \(T_E\) from a semilattice E provides an important tool in the study of inverse semigroups and ample semigroups. Munn’s semigroup \(T_E\) has the property that a semigroup is a fundamental inverse semigroup (resp. a fundamental ample semigroup) with a semilattice of idempotents isomorphic to E if and only if it is embeddable as a full inverse subsemigroup (resp. a full subsemigroup) into \(T_E\). The aim of this paper is to extend Munn’s approach to a class of abundant semigroups, namely abundant semigroups with a multiplicative ample transversal. We present here a semigroup \(T_{(I,\Lambda , E^{\circ }, P)}\) from a so-called admissible quadruple\((I,\Lambda , E^{\circ }, P)\) that plays for abundant semigroups with a multiplicative ample transversal the role that \(T_E\) plays for inverse semigroups and ample semigroups. More precisely, we show that a semigroup is a fundamental abundant semigroup (resp. fundamental regular semigroup) having a multiplicative ample transversal (resp. multiplicative inverse transversal) whose admissible quadruple is isomorphic to \((I,\Lambda , E^{\circ }, P)\) if and only if it is embeddable as a full subsemigroup (resp. full regular subsemigroup) into \(T_{(I,\Lambda , E^{\circ }, P)}\). Our results generalize and enrich some classical results of Munn on inverse semigroups and of Fountain on ample semigroups.

Regular subsemigroups of the semigroups of transformations preserving a fence

It is well-known that the transformation semigroup [Formula: see text] is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider [Formula: see text] as the base set of a fence [Formula: see text]. Two subsemigroups of [Formula: see text] are studied. Namely, the semigroup [Formula: see text] of all order-decreasing self-mappings of [Formula: see text] and the semigroup [Formula: see text] of all order-preserving self-mappings of [Formula: see text]. In this paper, we obtain that [Formula: see text] is a coregular subsemigroup of [Formula: see text]. A characterization of regular subsemigroups [Formula: see text] of [Formula: see text] is given, that is, [Formula: see text] is regular if and only if [Formula: see text]. Finally, we discuss the regularity of elements in [Formula: see text].