Algebra Universalis 

About: Algebra Universalis is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Variety (universal algebra) & Distributive lattice. It has an ISSN identifier of 0002-5240. Over the lifetime, 2796 publications have been published receiving 27202 citations.

The Birkhoff theorem for finite algebras

Abstract: A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms. It is well-known that explicit operations (those induced by τ-terms) do not suffice for such an equational description. Topological aspects of implicit operations are considered. Various examples are given.

Chain-complete posets and directed sets with applications

Abstract: Let a poset P be called chain-complete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets satisfying some sort of chain-completeness condition (see [,11, [-31, [6], I-71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the notion of chain-completeness and demonstrate its usefulness for various applications. Chain-complete posets behave in many respects like complete lattices; in fact, a chaincomplete lattice is a complete lattice. But in many cases it is the existence of sup's of chains, and not the existence of arbitrary sup's, that is crucial. More generally, let P be called chain s-complete when every chain of cardinality not greater than ~ has a sup. We first show that if a poset P is chain s-complete, then every directed subset of P with cardinality not exceeding ct has a sup in P. This sharpens the known result ([,8], [,181) that in any chain-complete poset, every directed set has a sup. Often a property holds for every directed set i f and only if it holds for every chain. We show that direct (inverse) limits exist in a category if and only if 'chain colimits' ('chain limits') exist. Since every chain has a well-ordered cofinal subset [11, p. 681, one need only work with well-ordered collections of objects in a category to establish or disprove the existence of direct and inverse limits. Similarly, a topological space is compact if and only if every 'chain of points' has a cluster point. A 'chain of points' is a generalization of a sequence. Chain-complete posers, like complete lattices, arise from closure operators in a fairly direct manner. Using closure operators we show how to form the chaincompletion P of any poset P. The chain-completion/~ of a poset P is a chain-complete poset with the property that any chain-continuous map from a poser P into a chain-complete poset Q extends uniquely to a chain-continuous map from the completion/~ into Q, where by a chaincontinuous map we mean one that preserves sup's of chains. If P is already chaincomplete, then/~ is naturally isomorphic to P. This completion is not the MacNeille

On the structure of hoops

Abstract: A hoop is a naturally ordered pocrim (ie, a partially ordered commutative residuated integral monoid) We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its finite members

Existence theorems for weakly symmetric operations

Abstract: A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this

Countable homogeneous partially ordered sets

Abstract: It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given.