In this short note, we describe all the elements in the semigroup

$$\begin{eqnarray}S(X,Y)=\{f\in {\mathcal{T}}_{X}:f(Y)\subseteq Y\}\end{eqnarray}$$

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where $\mathsf{D}(G)$ denotes the Davenport constant of $G$. The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a semigroup. Elements $a,b$ of $S$ are $\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then $S$ is weakly left ample if (1) idempotents of $S$ commute, (2) $\widetilde{\mathscr{R}}$ is a left congruence, (3) for any $a \in S$, $a$ is $\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say $a^+$, and (4) for any element $a$ and idempotent $e$ of $S$, $ae=(ae)^+a$. Elements $a,b$ of $S$ are $\mathscr{R}^*$-related if, for any $x,y \in S^1$, $xa=ya$ if and only if $xb=yb$. Then $S$ is left ample if it satisfies (1), (3) and (4) relative to $\mathscr{R}^*$ instead of $\widetilde{\mathscr{R}}$. Further, $S$ is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open.

In this paper, we characterise the quasi-strong endomorphisms of the join of split graphs. We give conditions under which the quasi-strong endomorphisms of the join of split graphs form a monoid.

In this paper we shall give characterizations of the closed subsemigroups of a Clifford semigroup. Also, we shall show that the class of all Clifford semigroups satisfies the strong isomorphism property and so is globally determined. Thus the results obtained by Kobayashi [‘Semilattices are globally determined’, Semigroup Forum29 (1984), 217–222] and by Gould and Iskra [‘Globally determined classes of semigroups’ Semigroup Forum28 (1984), 1–11] are generalized.

For a discrete abelian cancellative semigroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ with a weight function $\omega $ and associated multiplier semigroup $M_\omega (S)$ consisting of $\omega $-bounded multipliers, the multiplier algebra of the Beurling algebra of $(S,\omega )$ coincides with the Beurling algebra of $M_\omega (S)$ with the induced weight.

We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

In this paper we investigate some subclasses of strongly regular congruences on an $E$-inversive semigroup $S$. We describe the minimum and the maximum strongly orthodox congruences on $S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on $S$ with characteristic trace $\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an $E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence.

A semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.

For a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

We consider several familiar varieties of completely regular semigroups such as groups and completely simple semigroups. For each of them, we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties $ \mathcal{V} $ we determine minimal non-$ \mathcal{V} $ varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups. Three tables and a figure illustrate the results obtained.

Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

Towards an involutive analogue of a result on the semisimplicity of ${\ell }^{1} (S)$ by Hewitt and Zuckerman, we show that, given an abelian $\ast $-semigroup $S$, the commutative convolution Banach $\ast $-algebra ${\ell }^{1} (S)$ is $\ast $-semisimple if and only if Hermitian bounded semicharacters on $S$ separate the points of $S$; and we search for an intrinsic separation property on $S$ equivalent to $\ast $-semisimplicity. Very many natural involutive analogues of Hewitt and Zuckerman’s separation property are shown not to work, thereby exhibiting intricacies involved in analysis on $S$.

Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote

$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$

${T}_{\exists } (X)$

${ \mathcal{T} }_{X} $

${T}_{\exists } (X)$

For a finite Clifford inverse algebra $A$, with natural order meet-semilattice ${Y}_{A} $ and group of units ${G}_{A} $, we show that the inverse monoid obtained as the semidirect product ${ Y}_{A}^{1} {\mathop{\ast }\nolimits}_{\rho } {G}_{A} $ has a log-polynomial free spectrum whenever $\rho $ is a term-expressible left action of ${G}_{A} $ on ${Y}_{A} $ and all subgroups of $A$ are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.

For an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$. For $ \alpha \in \mathcal {I}(X)$, let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$. For an arbitrary $ \alpha \in \mathcal {I}(X)$, we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$, describe the regular elements of $C(\alpha )$, and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$, we determine the structure of $C(\alpha )$in terms of Green’s relations.

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.