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For a set X, denote by T(X) the semigroup of full transformations on X. For any subset Y of X and any subsemigroup $${\mathbb {S}}(Y)$$ of T(Y), denote by $$T_{{\mathbb {S}}(Y)}(X)$$ the semigroup of all transformations $$\alpha \in T(X)$$ such that $$\alpha |_{Y}\in {\mathbb {S}}(Y)$$ , where $$\alpha |_Y$$ is the restriction of $$\alpha $$ to Y. In this paper, we describe the regular elements of $$T_{{\mathbb {S}}(Y)}(X)$$ and determine when $$T_{{\mathbb {S}}(Y)}(X)$$ is a regular semigroup [inverse semigroup, completely regular semigroup]. With the assumption that $${\mathbb {S}}(Y)$$ contains the identity $${{\,\mathrm{id}\,}}_{{\tiny Y}}$$ , we describe Green’s relations in $$T_{{\mathbb {S}}(Y)}(X)$$ in terms of the corresponding Green’s relations in $${\mathbb {S}}(Y)$$ . We apply these general results to obtain more concrete results for the semigroup $$T_{\Gamma (Y)}(X)$$ , where $$\Gamma (Y)$$ is the semigroup of full injective transformations on Y. We also discuss generalizations and extensions of the semigroup $$T_{{\mathbb {S}}(Y)}(X)$$ .

This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with inverse semigroups, and then turn to an analysis of pseudogroups using sheaf theory. Our work includes some new material on wide semigroup homomorphisms and their geometric morphisms.

We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every , we have or . We obtain some properties that describe the behaviour of the Green’s equivalence relations , , and . Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup is presented.

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup is u – inverse semigroup. We will also prove that if (S,.) is a H - semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

For an admissible congruence $$\gamma $$ on an ample semigroup S, the minimum admissible congruence $$\gamma _t$$ whose trace is $$\mathrm {tr}\gamma $$ has been determined by the author, and Fountain, Gomes and Gould. With some restriction on $$\gamma $$ , we characterize in this paper the minimum admissible congruence $$(\gamma _t)_k$$ whose kernel is $$\mathrm {ker}\gamma _t$$ , as well as the minimum admissible congruence $$((\gamma _t)_k)_t$$ whose trace is $$\mathrm {tr}(\gamma _t)_k$$ . These results extend the corresponding results of Petrich and Reilly in the inverse semigroup case. Meanwhile, some particular minimum admissible congruences on S are recognized and this information is applied to set up a network of certain minimum admissible congruences on S.

We show that for any weakly cancellative uniformly locally finite inverse semigroup S such that $$l^{1}(S)$$ is ideally Connes amenable, for each $${\mathcal {D}}$$ -classes D of S, E(D) is finite, where E(D) is the set of idempotents of D. This is similar to a theorem of Duncan and Paterson, that says if $$l^{1}(S)$$ is amenable then E(D) is finite. Also we study ideal Connes amenability of $$l^{1}$$ -Munn algebras.

In this short note, we give some counter examples which show that [11, Proposition 3.5] is not true. As a consequence, the arguments in [11, Proposition 4.10] are not valid.

Demonic composition is defined on the set of binary relations over the non-empty set X, $$Rel_X$$ , and is a variant of standard or “angelic” composition. It arises naturally in the setting of the theory of non-deterministic computer programs, and shares many of the nice features of ordinary composition (it is associative, and generalises composition of functions). When equipped with the operations of demonic composition and domain, $$Rel_X$$ is a left restriction semigroup (like $$PT_X$$ , the semigroup of partial functions on X), whereas usual composition and domain give a unary semigroup satisfying weaker laws. By viewing $$Rel_X$$ under a restricted version of its usual composition and domain as a constellation (a kind of “one-sided” category), we show how this demonic left restriction semigroup structure arises on $$Rel_X$$ , placing it in a more general context. The construction applies to any unary semigroup with a “domain-like” operation satisfying certain minimal conditions which we identify. In particular it is shown that using the construction, any Baer $$*$$ -semigroup S can be given a left restriction semigroup structure which is even an inverse semigroup if S is $$*$$ -regular. It follows that the semigroup of $$n\times n$$ matrices over the real or complex numbers is an inverse semigroup with respect to a modified notion of product that almost always agrees with the usual matrix product, and in which inverse is pseudoinverse (Moore–Penrose inverse).

In this paper, in analogy with the classical case, we introduce module approximately biprojective and module approximately biflat Banach algebras. For an inverse semigroup S with the set of idempotents E, we show that the concepts of module approximate biprojectivity and module pseudo-amenability for $$l^1(S,\omega )$$ coincide (as an $$l^1(E)$$ -module). We also find necessary and sufficient conditions for the weighted semigroup algebra $$l^{1}(S,\omega )$$ to be module approximately biprojective and module approximately biflat. Finally, we study the module cohomological properties of projective tensor product of Banach algebras.

We obtain sufficient criteria for simplicity of systems, that is, rings $R$ that are equipped with a family of additive subgroups $R_s$, for $s \in S$, where $S$ is a semigroup, satisfying $R = \sum_{s \in S} R_s$ and $R_s R_t \subseteq R_{st}$, for $s,t \in S$. These criteria are specialized to obtain sufficient criteria for simplicity of, what we call, s-unital epsilon-strong systems, that is systems where $S$ is an inverse semigroup, $R$ is coherent, in the sense that for all $s,t \in S$ with $s \leq t$, the inclusion $R_s \subseteq R_t$ holds, and for each $s \in S$, the $R_s R_{s^*}$-$R_{s^*}R_s$-bimodule $R_s$ is s-unital. As an aplication of this, we obtain generalizations of recent criteria for simplicity of skew inverse semigroup rings, by Beuter, Goncalves, \"{O}inert and Royer, and then, in turn, for Steinberg algebras, over non-commutative rings, by Brown, Farthing, Sims, Steinberg, Clark and Edie-Michel.

In this paper we study submonoids of the monoid $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1802.03598; and text overlap with arXiv:1908.08521 by other authors

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right \'etale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

We introduce X-to-join representations of inverse semigroups, which are a relaxation of the notion of a cover-to-join representation. We construct the universal X-to-join Booleanization of an inverse semigroup S as a weakly meet-preserving quotient of the universal Booleanization \({\mathrm B}(S)\) and show that all such quotients of \({\mathrm B}(S)\) arise via X-to-join representations. As an application, we provide groupoid models for the intermediate boundary quotients of the \(C^*\)-algebra of a Zappa–Szep product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an [Formula: see text]-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying [Formula: see text]-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of [Formula: see text]-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an [Formula: see text]-unitary good cover.

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed.

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known., 43 pages

For an arbitrary set X and an equivalence relation $$\mu$$ on X, denote by $$P_\mu (X)$$ the semigroup of partial transformations $$\alpha$$ on X such that $$x\mu \subseteq x(\ker (\alpha ))$$ for every $$x\in {{\,\mathrm{dom}\,}}(\alpha )$$ , and the image of $$\alpha$$ is a partial transversal of $$\mu$$ . Every transversal K of $$\mu$$ defines a subgroup $$G=G_{K}$$ of $$P_\mu (X)$$ . We study subsemigroups $$\langle G,U\rangle$$ of $$P_\mu (X)$$ generated by $$G\cup U$$ , where U is any set of elements of $$P_\mu (X)$$ of rank less than $$|X/\mu |$$ . We show that each $$\langle G,U\rangle$$ is a regular semigroup, describe Green’s relations and ideals in $$\langle G,U\rangle$$ , and determine when $$\langle G,U\rangle$$ is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top $$\mathcal {J}$$ -class J of $$P_\mu (X)$$ is a right group. We find formulas for the ranks of the semigroups J, $$G\cup I$$ , $$J\cup I$$ , and I, where I is any proper ideal of $$P_\mu (X)$$ .

In this paper we initiate the study of $$\aleph _0$$ -categorical semigroups, where a countable semigroup S is $$\aleph _0$$ -categorical if, for any natural number n, the action of its group of automorphisms $${\text {Aut}}(S)$$ on $$S^n$$ has only finitely many orbits. We show that $$\aleph _0$$ -categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between $$\aleph _0$$ -categoricity and a number of semigroup and monoid constructions, namely Brandt semigroups, direct sums, 0-direct unions, semidirect products and $${\mathcal {P}}$$ -semigroups. As a corollary, we determine the $$ \aleph _0$$ -categoricity of an E-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let R be a commutative ring with an identity, and let S be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of S are all completely 0-simple. Then, we prove that the contracted semigroup algebra $$R_0[S]$$ is $$\pi $$ -semisimple if and only if the contracted semigroup algebras of all the principal factors of S are $$\pi $$ -semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of S cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.

We prove that if S is an E-solid locally inverse semigroup, and ρ is an inverse semigroup congruence on S such that the idempotent classes of p are completely simple semigroups thenSis embeddable into a λ-semidirectproduct of a completely simple semigroup byS/ρ. Consequently, theE-solid locally inverse semigroups turn out to be, up to isomorphism, the regularsubsemigroups of λ-semidirect products of completely simple semigroups by inverse semigroups

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of $n\times n$ upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of $2\times 2$ upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of "chain structured tropical matrix semigroups"; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by $3\times 3$ upper triangular matrix semigroups, and a new proof of the fact that this semigroup satisfies the same identities as the bicyclic monoid., Comment: 21 pages. This amended version of the author accepted manuscript contains a corrected proof of Proposition 7.1. The new proof establishes the proposition exactly as originally published, all other results are unaffected and the manuscript is otherwise unamended

Let G be a group. We show that the Birget–Rhodes prefix expansion $$G^{Pr}$$ and the Margolis–Meakin expansion M(X; f) of G with respect to $$f:X\rightarrow G$$ can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid $$U/\zeta $$ which is a homomorphic image of the free product U of the free semigroup $$X^+$$ on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with $$S/\sigma _S=G$$ , from which the characterizing universal properties of $$G^{Pr}$$ and M(X; f) can be recaptured easily.

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. They may be defined by a simple set of identities and the author initiated a study of the lattice of varieties of such semigroups, in parallel with the study of the lattice of varieties of two-sided restriction semigroups. In this work we study the subvariety $\mathbf{B}$ generated by Brandt semigroups and the subvarieties generated by the five-element Brandt inverse semigroup $B_{2}$, its four-element restriction subsemigroup $B_{0}$ and its three-element left restriction subsemigroup $D$. These have already been studied in the ‘plain’ semigroup context, in the inverse semigroup context (in the first two instances) and in the two-sided restriction semigroup context (in all but the last instance). The author has previously shown that in the last of these contexts, the behavior is pathological: ‘almost all’ finite restriction semigroups are inherently nonfinitely based. Here we show that this is not the case for left restriction semigroups, by exhibiting identities for the above varieties and for their joins with monoids (the analog of groups in this context). We do so by structural means involving subdirect decompositions into certain primitive semigroups. We also show that each identity has a simple structural interpretation.

In this paper we characterize the Johnson pseudo-contractibility of \(\ell ^{1}(S)\), where S is a uniformly locally finite inverse semigroup. We show that for a Brandt semigroup \(S=M^{0}(G,I)\) over a non-empty set I, \(\ell ^{1}(S)\) is Johnson pseudo-contractible if and only if G is amenable and I is finite. We give some examples to show the difference between Johnson pseudo-contractibility, pseudo-amenability and pseudo-contractibility among the semigroup algebras.

We study the Fourier–Stieltjes algebra of Roelcke precompact, non-archimedean, Polish groups and give a model-theoretic description of the Hilbert compactification of these groups. We characterize the family of such groups whose Fourier–Stieltjes algebra is dense in the algebra of weakly almost periodic functions: those are exactly the automorphism groups of ℵ 0 \aleph _0 -stable, ℵ 0 \aleph _0 -categorical structures. This analysis is then extended to all semitopological semigroup compactifications S S of such a group: S S is Hilbert-representable if and only if it is an inverse semigroup. We also show that every factor of the Hilbert compactification is Hilbert-representable.

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.

We define a class of inverse semigroup amalgams and derive normal forms for the amalgamated free products in the variety of semigroups. The class includes all amalgams of finite inverse semigroups, recently studied by Cherubini, Jajcayova, Meakin, Nuccio, Piochi and Rodaro (2005–2014), and lower bounded amalgams, that were introduced by the author (1997). We provide sufficient conditions for decidable word problem. We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups. The normal forms can be used to study amalgams in subvarieties of inverse semigroups. In a forthcoming paper by the author, the results are used for varieties of semilattices of groups.

We unify various étale groupoid reconstruction theorems such as the following:•Kumjian and Renault’s reconstruction from a groupoid C*-algebra;•Exel’s reconstruction from an ample inverse semigroup;•Steinberg’s reconstruction from a groupoid ring;•Choi, Gardella and Thiel’s reconstruction from a groupoidLp{L^{p}}-algebra.We do this by working with certainbumpysemigroupsSof functions defined on an étale groupoidG. The semigroup structure ofStogether with the diagonal subsemigroupDthen yields a natural domination relation≺{\prec}onS. The groupoid of≺{\prec}-ultrafilters is then isomorphic to the original groupoidG.

For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$ Our main result is that $M(X)$ is an inverse semigroup, therefore, one can define the $C^*$-algebra of this inverse semigroup. We characterize the metrics that are idempotents, find a minimal projection in $M(X)$ and give examples of metric spaces, for which the semigroup $M(X)$ is commutative. We show that if the Gromov-Hausdorff distance between two metric spaces, $X$ and $Y$, is finite then $M(X)$ and $M(Y)$ are isomorphic. We also describe the class of metrics determined by subsets of $X$ in terms of the closures of the subsets in the Higson corona of $X$., 15 pages, final version, to appear in J. Geom. Anal

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Sch\"{u}tzenberger graphs. This graph, which we denote by $\Lambda_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(\Lambda_S)$. This yields a chacterization of when $\mathcal{R}_S = C^*_u(\Lambda_S)$, which generalizes finite generation of $S$. We have termed this by finite labeability (FL), since it holds when the $\Lambda_S$ can be labeled in a finitary manner. The graph $\Lambda_S$, and the FL condition above, also allow to analyze large scale properties of $\Lambda_S$ and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of $S$ (a notion generalizing Day's definition of amenability of a semigroup, cf., [5]) is a quasi-isometric invariant of $\Lambda_S$. Moreover, we characterize property A of $\Lambda_S$ (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view., Comment: Accepted version for publication; minor changes

We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every infinite cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup., Comment: 10 pages

In this paper we study the structure of the C*-algebra, generated by the representation of the paths semigroup on a partially ordered set (poset) and get the net of isomorphic C*-algebras over this poset. We construct the extensions of this algebra, such that the algebra is an ideal in that extensions and quotient algebras are isomorphic to the Cuntz algebra.

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R0[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.

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.

A fence is a particular partial order on a (finite) set, close to the linear order. In this paper, we calculate the rank of the semigroup $$\mathscr {FI}_{n}$$ of all order-preserving partial injections on an n-element fence. In particular, we provide a minimal generating set for $$\mathscr {FI}_{n}$$ . In the present paper, n is odd since this problem for even n was already solved by I. Dimitrova and J. Koppitz.

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

In 1959, P. Hall introduced the locally finite group $\mathcal{U}$, today known as Hall's universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in $\mathcal{U}$. It can be explicitly described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraisse limit of the class of all finite groups. Since its introduction Hall's group, and several natural generalisations, have been widely studied. In this article we use a generalisation of Fraisse theory to construct a countable, universal, locally finite semigroup $\mathcal{T}$, that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup $\mathcal{I}$ which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups $\mathcal{T}$ and $\mathcal{I}$ are the natural counterparts of Hall's universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall's group itself., 25 pages, 2 figures; to appear in Forum Mathematicum

Let ℐn be the symmetric inverse semigroup on Xn = {1, 2,…, n}, and let 𝒟𝒫n and 𝒪𝒟𝒫n be its subsemigroups of partial isometries and of order-preserving partial isometries of Xn, respectively. In this article, we investigate the cycle structure of a partial isometry and characterize the Green's relations on 𝒟𝒫n and 𝒪𝒟𝒫n. We show that 𝒪𝒟𝒫n is a 0-E-unitary inverse semigroup. We also investigate the ranks of 𝒟𝒫n and 𝒪𝒟𝒫n.

In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra \(\ell _1(S,\omega )\) implies the countability of S. This generalizes the main result of Craw and Young (Q J Math 25:351–358, 1974), stating that every countable semigroup admits a weight \(\omega \) for which \(\ell _1(S,\omega )\) is Arens regular and no uncountable group admits such a weight.

We prove that if S is a commutative semigroup with well founded universal semilattice or a solvable inverse semigroup with well founded semilattice of idempotents, then every strongly productive ultrafilter on S is idempotent. Moreover we show that any very strongly productive ultrafilter on the free semigroup with countably many generators is sparse, answering a question of Hindman and Legette Jones., Comment: 14 pages, accepted for publication in "Semigroup Forum"

We show that for an inverse semigroup S with the set idempotents E acting on S trivially from left and by multiplication from right, any bounded module derivation from \(\ell ^1(S)\) to \(({\ell ^1(S)}/{J})^*=J^{\perp }\) is inner, where J is the closed ideal generated by elements of the form \(\delta _{set}-\delta _{st}\) with \(s,t\in S\) and \(e\in E\).