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In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups., 27 pages

A congruence on an inverse semigroup S is determined uniquely by its kernel and trace. Denoting by $$\rho _k$$ and $$\rho _t$$ the least congruence on S having the same kernel and the same trace as $$\rho$$ , respectively, and denoting by $$\omega$$ the universal congruence on S, we consider the sequence $$\omega$$ , $$\omega _k$$ , $$\omega _t$$ , $$(\omega _k)_t$$ , $$(\omega _t)_k$$ , $$((\omega _k)_t)_k$$ , $$((\omega _t)_k)_t$$ , $$\ldots$$ . The quotients $$\{S/\omega _k\}$$ , $$\{S/\omega _t\}$$ , $$\{S/(\omega _k)_t\}$$ , $$\{S/(\omega _t)_k\}$$ , $$\{S/((\omega _k)_t)_k\}$$ , $$\{S/((\omega _t)_k)_t\}$$ , $$\ldots$$ , as S runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.

As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case., Comment: 14 pages including Appendix

In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an $F$-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition.

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an inner inverse, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. \v{S}ter, in a paper to appear, show that a much larger class of elements $x$ of rings $R,$ including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result proved is a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to belong to an inverse submonoid of the monoid of all endomaps of $S.$, Comment: 18pp. The main change from the preceding version is the discussion of three questions posed by the referee, two on p.10, starting on line 6, and one starting at the top of p.16. There are also many small revisions of wording etc

In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a\sim_{\mathrm{i}} b$, if there exists $g\in S^1$ such that $g^{-1} ag=b$ and $gbg^{-1} =a$. The purpose of this paper is to study the conjugacy $\sim_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister $P$-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups., Comment: 22 pages

Although v3 should have the same numberings as the published version, the page numbers may not be the same.ERRATA (so far harmless):Line -4 of proof of Lemma 3.1.3: \lambda_a preserves all joins from S\downarrow d(a) (not S). Also, formulation of Lemma 3.1.3 somewhat suboptimal (write \lambda_a : a^{-1}S\to aS, etc.)Lemma 3.1.4: the preservation of all joins by d and r is OK, but not the one of meets (compatibility needed).Lemma 6.7.6: the canonical map f should go from Typ \kappa(G) to V(k(\kappa(G))); International audience; For an action of a group G on a set Ω, preserving a ring B of subsets of Ω, the commutative monoid freely generated by elements [X], for X ∈ B, subjected to the relations [∅] = 0, [gX] = [X] (for g ∈ G), and [X Y ] = [X] + [Y ] (where denotes disjoint union), is called the monoid of equidecomposability types of elements of B, with respect to G, and denoted by Z + B/ /G. It is well known that Z + B/ /G is a conical refinement monoid. We observe, as an easy consequence of known results, that every countable conical refinement monoid appears as Z + B/ /G, and we develop the underlying algebraic theory, discussing in detail the quotients of refinement monoids by special sorts of congruences called V-congruences. Having in mind representation problems in nonstable K-theory of rings and operator algebras, we are naturally led to type monoids of Boolean inverse semigroups. Observing that those monoids are identical to monoids of equi-decomposability types, and formally similar to those appearing in nonstable K-theory of von Neumann regular rings, we investigate various similarities and differences between those theories. In the process, we prove that Boolean inverse semigroups form a congruence -permutable variety in the sense of universal algebra. We deduce from this that they encode a large number of embedding problems of (not necessarily Boolean) inverse semigroups into involutary rings and C*-algebras.

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property., Comment: 6 pages

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We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff \'etale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick does not generalise to non-Hausdorff groupoids., Comment: 59 pages. Proofreading with minor changes. Version accepted for publication at Rocky Mountain Journal