Anna B. Romanowska

Duality for distributive bisemilattices

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

Bisemilattices of subsemilattices

Abstract Sets of subsemilattices of semilattices are given a natural meet-distributive bisemilattice structure. These bisemilattices are decomposed into constituent semilattices and distributive lattices. Their construction furnishes a left adjoint to the forgetful functor from meet-distributive bisemilattices to semilattices. Representation theorems for meet-distributive bisemilattices follow.

Duality for semilattice representations

Abstract The paper presents general machinery for extending a duality between complete, cocomplete categories to a duality between corresponding categories of semilattice representations (i.e. sheaves over Alexandrov spaces). This enables known dualities to be regularized. Among the applications, regularized Lindenbaum-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of PASCAL-like programming languages) is the logic for semilattice-indexed systems of sets. Another application enlarges Pontryagin duality by regularizing it to obtain duality for commutative inverse Clifford monoids.

Semilattice-based dualities

The paper discusses “regularisation” of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities.

On the structure of subalgebra systems of idempotent entropic algebras

A general study of the subalgebra systems of idempotent entropic algebras was begun in [21], where these systems were considered as abstract algebras called idempotent-entropic operatoror IEO-semilattices. The current paper examines the structure of these algebras by means of various decompositions of them, together with corresponding construction methods for recovering the algebras from their decompositions. One of the basic construction methods in semigroup theory is A. G. Clifford’s “strong semilattice of semigroups,” generalised to other kinds of algebra without nullary operations by J. Plonka. This method is now referred to as the “Plonka sum.” It arises from a simultaneous consideration of semilattices both as abstract algebras of general type and as categories with finite products. The method is recalled in Section 2, which then goes on to introduce a generalisation of the Plonka sum required in Section 5. Some of the properties of the generalised Plonka sum are

Quasivarieties of Cancellative Commutative Binary Modes

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

Subquasivarieties of regularized varieties

This paper considers the lattice of subquasivarieties of a regular variety. In particular we show that if V is a strongly irregular variety that is minimal as a quasivariety, then the smallest quasivariety containing both V and SI (the variety of semilattices) is never equal to the regularization V of V.We use this result to describe the lattice of subquasivarieties of V in several special but quite common, cases and give a number of applications and examples.

Varieties of Birkhoff Systems Part II

This is the second part of a two-part paper on Birkhoff systems. A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). The first part of this paper described the lattice of subvarieties of Birkhoff systems. This second part continues the investigation of subvarieties of Birkhoff systems. The 4-element subdirectly irreducible Birkhoff systems are described, and the varieties they generate are placed in the lattice of subvarieties. The poset of varieties generated by finite splitting bichains is described. Finally, a structure theorem is given for one of the five covers of the variety of distributive Birkhoff systems, the only cover that previously had no structure theorem. This structure theorem is used to complete results from the first part of this paper describing the lower part of the lattice of subvarieties of Birkhoff systems.

ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

Abstract. Comonoid, bi-algebra, and Hopf algebra structures are stud-ied within the universal-algebraic context of entropic varieties. Attentionfocuses on the behavior of setlike and primitive elements. It is shown thatentropic Jo´nsson-Tarski varieties provide a natural universal-algebraicsetting for primitive elements and group quantum couples (generaliza-tions of the group quantum double). Here, the set of primitive elementsof a Hopf algebra forms a Lie algebra, and the tensor algebra on anyalgebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then theunderlying Jo´nsson-Tarski monoid of the generating algebra is cancella-tive. The problem of determining when the Jo´nsson-Tarski monoid formsa group is open. 1. IntroductionThe aim of this paper is to consider comonoid, bi-algebra, and Hopf alge-bra structure within the context of universal algebra. The general categoricalsetting for such structure is provided by symmetric monoidal categories (or“symmetric tensor categories”, compare [11, p. 69]), with an associative tensorproduct and a unit object. Now an algebra (A,Ω) of type τ : Ω → Nis said tobe entropic if the operationω: A

DUALITY FOR CONVEX POLYTOPES

This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. 2000 Mathematics subject classification: primary 18C05, 52A01; secondary 08A35, 08A05, 18A35.