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Given a semigroup $S$, for each Green's relation $\mathcal{K}\in\{\mathcal{L},\mathcal{R},\mathcal{J},\mathcal{H}\}$ on $S,$ the $\mathcal{K}$-height of $S,$ denoted by $H_{\mathcal{K}}(S),$ is the height of the poset of $\mathcal{K}$-classes of $S.$ More precisely, if there is a finite bound on the sizes of chains of $\mathcal{K}$-classes of $S,$ then $H_{\mathcal{K}}(S)$ is defined as the maximum size of such a chain; otherwise, we say that $S$ has infinite $\mathcal{K}$-height. We discuss the relationships between these four $\mathcal{K}$-heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite $\mathcal{L}$-height if and only if it has finite $\mathcal{R}$-height if and only if it has finite $\mathcal{J}$-height. In fact, for a stable semigroup $S,$ if $H_{\mathcal{L}}(S)=n$ then $H_{\mathcal{R}}(S)\leq2^n-1$ and $H_{\mathcal{J}}(S)\leq2^n-1,$ and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if $2\leq H_{\mathcal{L}}(S)<\infty$ and $2\leq H_{\mathcal{R}}(S)<\infty,$ then $H_{\mathcal{J}}(S)\leq H_{\mathcal{L}}(S)+H_{\mathcal{R}}(S)-2.$ We also show that for each $n\in\mathbb{N}$ there exists a semigroup $S$ such that $H_{\mathcal{L}}(S)=H_{\mathcal{R}}(S)=2^n+n-3$ and $H_{\mathcal{J}}(S)=2^{n+1}-4.$ By way of contrast, we prove that for a regular semigroup the $\mathcal{L}$-, $\mathcal{R}$- and $\mathcal{H}$-heights coincide with each other, and are greater or equal to the $\mathcal{J}$-height. Moreover, in a stable, regular semigroup the $\mathcal{L}$-, $\mathcal{R}$-, $\mathcal{H}$- and $\mathcal{J}$-heights are all equal.

For a semigroup $S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right-$FP_1$), the right diameter of $S$ is a parameter that expresses how `far apart' elements of $S$ can be from each other, in a certain sense. To be more precise, for each finite generating set $U$ for the universal right congruence on $S,$ we have a metric space $(S,d_U)$ where $d_U(a,b)$ is the minimum length of derivations for $(a,b)$ as a consequence of pairs in $U$; the right diameter of $S$ with respect to $U$ is the diameter of this metric space. The right diameter of $S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We investigate whether various natural infinite semigroups of transformations and partitions have a finitely generated universal right/left congruence, and for those that do, we determine their right/left diameter. Among other results, for an arbitrary infinite set $X$ we prove the following. Each of the monoids of all binary relations on $X,$ of all partial transformations on $X,$ and of all full transformations on $X,$ as well as the partition and partial Brauer monoids on $X,$ have right diameter 1 and left diameter 1. The symmetric inverse monoid on $X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on $X$ has right diameter 4, and its minimal ideal (called the Baer-Levi semigroup on $X$) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on $X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

We call a semigroup $\mathcal{R}$-noetherian if it satisfies the ascending chain condition on principal right ideals, or, equivalently, the ascending chain condition on $\mathcal{R}$-classes. We investigate the behaviour of the property of being $\mathcal{R}$-noetherian under the following standard semigroup-theoretic constructions: semidirect products, Sch\"utzenberger products, free products, Rees matrix semigroups, Brandt extensions, Bruck-Reilly extensions and semilattices of semigroups.

We call a semigroup $S$ right noetherian if it satisfies the ascending chain condition on right ideals, and we say that $S$ satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.

A semigroup $S$ is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair $(s,t)\in S\times S$ as a consequence of those in $U$. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders $\leq_{\mathcal{L}}$ or $\leq_{\mathcal{J}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or $\mathcal{J}$-triviality.

The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we establish bounds on the $\mathcal{R}$-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite $\mathcal{R}$-height. We then investigate whether these bounds can be attained.

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterise weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterise weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal{H}$ is finitely generated.

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Sch\"utzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Sch\"utzenberger groups have the property., Comment: 27 pages

We call a semigroup $S$ f-noetherian if every right congruence of finite index on $S$ is finitely generated. We prove that every finitely generated semigroup is f-noetherian, and investigate whether the properties of being f-noetherian and being finitely generated coincide for various semigroup classes.

A semigroup $S$ is right noetherian if every right congruence on $S$ is finitely generated. In this paper we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.

We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid $M$ is said to preserve property $\mathcal{P}$ in direct products if, for any two $M$-acts $A$ and $B$, the direct product $A\times B$ has property $\mathcal{P}$ if and only if both $A$ and $B$ have property $\mathcal{P}$. It is proved that the monoids $M$ that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal $M$-act $M\times M$ is finitely generated (resp. finitely presented). We show that a wreath product $A\wr B$ is finitely generated if and only if both $A$ and $B$ are finitely generated. It is also proved that a necessary condition for $A\wr B$ to be finitely presented is that both $A$ and $B$ are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented., Comment: arXiv admin note: text overlap with arXiv:1709.08916

The purpose of this paper is to introduce the theory of presentations of monoids acts. We aim to construct `nice' general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an $M$-act $A$ and a subact $B$ of $A$, on the one hand we construct presentations for $B$ and the Rees quotient $A/B$ using a presentation for $A$, and on the other hand we derive a presentation for $A$ from presentations for $B$ and $A/B$. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact $B$ has finite complement in an $M$-act $A$. We show that if $M$ is a finitely generated monoid and $B$ is finitely presented, then $A$ is finitely presented. We also show that if $M$ belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.