Abstract: In his landmark paper on amalgamation published in Algebra Universalis in 1971, Don Pigozzi posed some open questions in connection with amalgamation of subclasses of cylindric algebras. Some of these questions were originally raised by Comer, Daigneault, Johnson, McKenzie and others. In this paper we give answers to all these as well as a number of other related questions. Most of the solutions were found by the authors of this paper. However, a few were contributed by others who will of course be given due credit at the appropriate points.

Abstract: We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples.

Abstract: In this paper we show that every quasi-projective relation algebra (QRA) has within it cylindric algebras (CAs) of all finite dimensions greater than two such that (i) they are term-definable in the QRA, (ii) they have a relation algebraic reduct that is isomorphic to the QRA and (iii) these isomorphisms are also term-definable. Furthermore, these CAs form a sequence such that each of them is isomorphic to a neat reduct of the higher dimensional ones, and these isomorphisms are also term-definable.

Abstract: We investigate pure monomorphisms and pure epimorphisms in varieties of universal algebras with applications to equationally compact (= pure injective) algebras, pure projective algebras and perfect varieties.

Abstract: We show that no finite set of first-order axioms can define the class of representable semilattice-ordered monoids.

Abstract: We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem.

Abstract: A tower in a Boolean algebra A is a strictly increasing sequence, of regular length, of elements of A with supremum 1. We consider the following functions:

Abstract: In 1962, the author proved that each reduced archimedean f-ring can be represented as an f-ring of continuous extended real-valued function on a locally compact space. The existence of this representation has proved to be quite useful; however, the proof so obscured the definition of the representing functions that deeper applications have remained out of reach. In this paper, we give a new proof of this result; one in which the derivation of the representing functions is more readily accessible. This accessibility is exploited to prove that f-ring homomorphisms on reduced archimedean f-rings are induced by continuous maps between subspaces of their representing spaces, and this leads to further insights into the structure of such homomorphisms.

Abstract: Is every full duality strong? We show that the answer is ‘no’, thereby answering a question that goes back to the earliest foundations of the theory of natural dualities.

Abstract: We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term.

Abstract: In this paper we investigate a particular subclass of pseudocomplemented Ockham algebras $$(L; \wedge,\vee,f,^{*}, 0, 1)$$ where $$(L; \wedge,\vee, f, 0, 1)$$ is an Ockham algebra, $$(L; \wedge, \vee,^{*} , 0, 1)$$ is a p-algebra, and the operations $$x \longmapsto f(x)$$ and $$x \longmapsto x^{*}$$ satisfy the identities $$f(x^{*}) = x^{**}$$ and $$[f(x)]^{*} = f^{2}(x)$$ . We show that there are precisely eleven non-isomorphic subdirectly irreducible members in the class of these algebras and give a complete description of them.

Abstract: ΠMTL-algebras were introduced as an algebraic counterpart of the cancellative extension of monoidal t-norm based logic. It was shown that they form a variety generated by ΠMTL-chains on the real interval [0, 1]. In this paper the structure of these generators is investigated. The results illuminate the structure of cancellative integral commutative residuated chains, because every such algebra belongs to the quasivariety generated by the zero-free subreducts on (0, 1] of all ΠMTL-chains on [0, 1].

Abstract: We study MV-relation-algebras, appearing by abstracting away from the concrete many-valued relations and the operations on them, such as composition and converse. MV-relation-algebras are MV generalizations of the relation algebras developed by A. Tarski and his school starting from the late forties. Some facts about ideals, congruences, and various types of elements are proved. A characterization of the ?natural? MV-relation-algebras (a parameterized analogue of the classical full proper relation algebras) is also provided, as well as a first-order elementary description of matrix MV-relation algebras.

Abstract: A ternary term m(x, y, z) of an algebra is called a majority term if the algebra satisfies the identities m(x, x, y) = x, m(x, y, x) = x and m(y, x, x) = x. A congruence α of a finite algebra is called uniform if all of its blocks (i.e., classes) have the same number of elements. In particular, if all the α-blocks are two-element then α is said to be a 2-uniform congruence. If all congruences of A are uniform then A is said to be a uniform algebra. Answering a problem raised by Gratzer, Quackenbush and Schmidt [2], Kaarli [3] has recently proved that uniform finite lattices are congruence permutable.

Abstract: We show how to alter the material in [4] to prove that every variety of modular ortholattices is generated by its simple members.

Abstract: A ring is clean if every element is the sum of a unit and an idempotent. Let $$ 1 \in A $$ be a dense local subring of the reals which is not a field. We show that the ring of A-valued continuous functions on a zero-dimensional space X is clean if and only if X is a P-space, and examine some properties of the prime ideal spectrum of this ring.

Abstract: The realm of natural dualities that are known to be full but not strong at the finite level is a very small one, consisting of a single example. This example, based on the three-element bounded distributive lattice, was presented by Davey, Haviar and Willard [8]. In this paper, we extend this realm to the class of all natural dualities based on a finite non-boolean bounded distributive lattice.

Abstract: Let A be a finite algebra and $${\mathcal{Q}}$$ a quasivariety. By $${\rm Con}_{{\mathcal{Q}}}$$ A is meant the lattice of congruences θ on A with $$A/\theta \in {\mathcal{Q}}$$ . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety $${\mathcal{R}}$$ such that $${\rm Con}_{R} A \cong L$$ .

Abstract: In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments \( \{ z \in V:a < z < b\} \), a < b, is a base for the Euclidean topology).

Abstract: A classical result about Boolean algebras independently proved by Magill [10], Maxson [11], and Schein [17] says that non-trivial Boolean algebras are isomorphic whenever their endomorphism monoids are isomorphic. The main point of this note is to show that the finite part of this classical result is true within monadic Boolean algebras. By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadic Boolean algebras the endomorphism monoid of each of which has only one element (namely, the identity), this being the first known example of a variety that is not universal (in the sense of Hedrlin and Pultr), but contains a proper class of non-isomorphic rigid algebras (that is, the identity is the only endomorphism).

Abstract: Let \(\underline{\text{M}}\) be a finite algebra. We show that if there exists a particular dualising alter ego \({\mathop{{\rm M}}\limits_{\sim}}^{\flat}\) that satisfies a weak form of injectivity, then the notions of full duality and strong duality for \(\underline{\text{M}}\) are equivalent.

Abstract: Given two complete atomistic lattices $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ , we define a set $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We call the elements of $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ weak tensor products of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ . We prove that $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ is a complete lattice. We compare the bottom element $${\mathcal{L}}_{1} $$ $$ {\mathcal{L}}_{2}$$ with the separated product of Aerts and with the box product of Gratzer and Wehrung. Similarly, we compare the top element $${\mathcal{L}}_{1} $$ $$ {\mathcal{L}}_{2}$$ with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ (true for instance if $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ .

Abstract: We characterize the congruence permutable MS-algebras in a result analogous to that of Nachbin for distributive lattices. In our main result we prove, among other things, that the congruence permutability of an MS-algebra L is equivalent to the existence of two binary operations in L which satisfy certain equational properties. We show that the class of all congruence permutable MS-algebras with these additional operations forms an equational class. Finally we give a Mal’cev term for this new variety.

Abstract: MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set \(\{b_i \mid i \in X\}\) for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then \(\bigwedge_{b_i\in A}\,b_i\,\wedge\,\,\bigwedge_{b_{j}\in B} eg b_j eq 0\). Our aim in this note is to give a similar characterization of free generating sets in MV-algebras.

Abstract: In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras. We show here that most results of [1] are, in fact, immediate consequences of two well-known facts: (i) a bounded commutative BCK-algebra possesses a natural upper semilattice structure, (ii) the class of MV-algebras and that of bounded commutative BCK-algebras are definitionally equivalent.

Abstract: The implicational subreducts of n-potent commutative integral residuated lattices are axiomatized using a new embedding of a BCK-algebra into a commutative integral residuated lattice. The class of {→, 1, ≤ }-subreducts of commutative residuated lattices satisfying xn ≤ xm is also axiomatized, as are other subreduct classes.

Abstract: We describe the clones on 3 elements that can be expressed as Pol ρ for ρ a binary relation. We present the poset of these clones ordered by inclusion. This article is a shortened version of the author’s thesis, to give an idea of the whole work.

Abstract: In this paper we investigate a class of inverse transformation monoids constructed from finite lattices, and we describe a necessary and sufficient condition for such a transformation monoid to be collapsing.