For every group G, the set $\mathcal {P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup $ and element-wise multiplication $\cdot $ , and forms an involution semigroup under $\cdot $ and element-wise inversion ${}^{-1}$ . We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring $(\mathcal {P}(G),\cup ,\cdot )$ nor the involution semigroup $(\mathcal {P}(G),\cdot ,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.

We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

The ring $\mathbb Z_{d}$ of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree $T_{d}$ . Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl. 4(2) (2012), 151–160] showed that $f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of $\mathbb Z_{p}\cap \mathbb Q$ . We generalize this result to arbitrary integers $d\geq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$ . We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$ .

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.

We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

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 characterize 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 characterize 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 Schützenberger 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 Schützenberger groups have the property.

For every pseudovariety $\mathbf {V}$ of finite monoids, let $\mathbf {LV}$ denote the pseudovariety of all finite semigroups all of whose local submonoids belong to $\mathbf {V}$ . In this paper, it is shown that, for every nontrivial semidirectly closed pseudovariety $\mathbf {V}$ of finite monoids, the pseudovariety $\mathbf {LV}$ of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety $\mathbf {V}$ is local in the sense of Tilson. This finding resolves a long-standing open problem posed in the second volume of the classic monograph by Eilenberg.

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept for considering quotients of inverse semigroups. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization.

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.

It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

If $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and $H$ is the subgroup of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\unicode[STIX]{x1D6E4},v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover $g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.

Subshifts with property $(A)$ are constructed from a class of directed graphs. As special cases the Markov–Dyck shifts are shown to have property $(A)$. The semigroups that are associated to ${\mathcal{R}}$-graph shifts with Property $(A)$ are determined.