Volume 48, Issue 4 - Special issue title: PROCEEDINGS OF THE AAA88 - 88TH WORKSHOP ON GENERAL ALGEBRA, Issue editors: Anna Romanowska (Warsaw University of Technology, Poland), Jonathan D.H. Smith (Iowa State University of Science and Technology, Ames, Iowa, USA)

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke-Kiselman monoids including the Kiselman monoids K n . As a consequence, we conclude that the identities of K n are nonfinitely based for every n ≥4 and exhibit a finite identity basis for the identities of each of the monoids K 2 and K 3 .

A class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset (R, ≤) is relatively orthocomplemented; in particular, every initial segment in it is orthomodular. The order ≤ is actually a version of the so called right-star order. The one-sided star orders are well-investigated for matrices and recently have been generalized to bounded linear Hilbert space operators and to abstract Rickart *-rings. The paper demonstrates that they can successfully be treated also in Rickart rings without involution.

In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear.

The paper deals with the monounary algebras for which the second centralizer equals the first centralizer. We describe Green’s relations on the semigroup C, where C is the centralizer of such algebra.

Let K be any field and L be any lattice. In this note we show that L is a sublattice of annihilators in an associative and commutative K-algebra. If L is finite, then our algebra will be finite dimensional over K.

We consider here some notions and results about sets of generators of finite algebras motivated by the case of finite groups. We illustrate these notions by simple examples closed to lattices and semigroups. Next, we examine these notions in the case of groups with operators.

In this paper, we characterize retracts of a wide class of Fraïssé limits using the tools developed in a recent paper by W. Kubis and the present author, which we refer to as Katetov functors. This approach enables us to conclude that in many cases, a structure is a retract of a Fraïssé limit if and only if it is algebraically closed in the surrounding category.

The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper.

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the “quantum” version of quasigroups, so one-sided quantum quasigroups are the “quantum” version of left or right quasigroups.

We take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.