In a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

Algebras (A, ∧, ∨, ~, γ, 0, 1) of type (2,2,1,1,0,0) such that (A, ∧, ∨, ~, γ 0, 1) is a De Morgan algebra and γ is a lattice homomorphism from A into its center that satisfies one of the conditions (i) a ≤ γa or (ii) a ≤ ~ a ∧ γa are considered. The dual categories and the lattice of their subvarieties are determined, and applications to Lukasiewicz algebras are given.

Starting with a class ℳ of Ω-groups, necessary and sufficient conditions on ℳ are given to ensure that the corresponding Hoehnke radical ρ (determined by the subdirect closure of ℳ as semisimple class) is a radical in the sense of Kurosh and Amitsur; has a hereditary semisimple class; satisfies the ADS-property; has a hereditary radical class or satisfies ρN ∩ I ⊆ ρI and lastly, have both a hereditary radical and semisimple class or satisfies ρN ∩ I = ρI.

In this final contribution to the investigation of commutator laws in groups, we answer some of the questions left open in the previous two papers. The principal result is the independence of the Jacobi-Witt-Hall type laws from the so-called standard set of laws. The main results of the earlier papers are summarised. An interlude corrects some of the numerous printing errors in the second of our papers.

It is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.

We prove that every locally finite, congruence modular, minimal variety is minimal as a quasivariety. We also construct all finite, strictly simple algebras generating a congruence distributive variety, such that the sett of unary term perations forms a group. Lastly, these results are applied to a problem in algebraic logic to give a sufficient condition for a deductive system to be structurally complete.

The main theorem of this paper shows that the lattice of congruences contained is some equivalence π on a semigroup S can be decomposed into a subdirect product of sublattices of the congruence lattices on the ‘prinipal π-facotrsρ of S—the semigroups formed by adjoining zeroes to the π-classes—whenever these are well-defined. The theorem is then applied to various equavalences and classes of semigroups to give some new results and alternative proofs of known ones.

We consider a variety of algebras with two binary commutative and associative operations. For each integer n ≥ 0, we represent the partitions on an n-element set as n-ary terms in the variety. We determine necessary and sufficient conditions on the variety ensuring that, for each n, these representing terms be all the essentially n-ary terms and moreover that distinct partitions yield distinct terms.

Given a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

In this paper we investigate subtractive varieties of algebras that are congruence quasi-orderable. Though this concept has its origin in abstract algebraic logic, it seems to be worth investigating in a purely algebraic fashion. Besides clarifying the algebraic meaning of this notion, we obtain several structure theorems about such varieties. Also several examples are provided to illustrate the theory.

An inverse semigroup S is said to be meet (join) semidistributive if its lattice (S) of full inverse subsemigroups is meet (join) semidistributive. We show that every meet (join) semidistributive inverse semigroup is in fact distributive.

The structure of semigroups with atomistic congruence lattices (that is, each congruence is the supremum of the atoms it contains) is studied. For the weakly reductive case the problem of describing the structure of such semigroups is solved up to simple and congruence free semigroups, respectively. As applications, all commutative, finite, completely semisimple semigroups, respectively, with atomistic congruence lattices are described.

Consider the quasi-variety generated by a finite algebra and assume that yields a natural duality on based on which is optimal modulo endomorphisms. We shoe that, provided satisfies certain minimality conditions, we can transfer this duality to a natural duality on based on , which is also optimal modulo endormorphisms, for any finite algebra in that has a subalgebra isomorphic to .

We show that mono-unary algebras have rank at most two and are thus strongly dualizable. We provide an example of a strong duality for a mono-unary algebra using an alter ego with (partial) operations of arity at most two. This mono-unary algebra has rank two and generates the same quasivariety as an injective, hence rank one, mono-unary algebra.

The free product *CRSi of an arbitrary family of disjoint completely simple semigroups {Si}i∈i, within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Kaďourek and Polák for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of *CRSi are free, except for those in the factors Si themselves. The general theorem simplifies in the case of free CR-products of groups and, in particular, free idempotent-generated completely regular semigroups.

In this paper we show that a bialgebraic context which arises from a duality in a fairly general way must arise from a duality between categories of modules. To show this, we give an elementary proof of Mitchell's Embedding Theorem for prevarieties.

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

The optimality of dualities on a quasivariety , generated by a finite algebra , has been introduced by Davey and Priestley in the 1990s. Since every optimal duality is determined by a transversal of a certain family of subsets of Ω, where Ω is a given set of relations yielding a duality on , an understanding of the structures of these subsets—known as globally minimal failsets—was required. A complete description of globally minimal failsets which do not contain partial endomorphisms has recently been given by the author and H. A. Priestley. Here we are concerned with globally minimal failsets containing endomorphisms. We aim to explain what seems to be a pattern in the way endomorphisms belong to these failsets. This paper also gives a complete description of globally minimal failsets whose minimal elements are automorphisms, when is a subdirectly irreducible lattice-structured algebra.

Pseudo-BL algebras are noncommutative generalizations of BL-algebras and they include pseudo-MV algebras, a class of structures that are categorically equivalent to l-groups with strong unit. In this paper we characterize directly indecomposable pseudo-BL algebras and we define and study different classes of these structures: local, good, perfect, peculiar, and (strongly) bipartite pseudo-BL algebras.

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.