Jaroslav Ježek

Definability in the lattice of equational theories of semigroups

We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, Tϖ} is definable, whereTϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable.

The equational theory of paramedial cancellation groupoids

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by paramedial cancellation groupoids. (By a cancellation groupoid we mean a groupoid satisfying both xz = yz → x = y and zx = zy → x = y.) Clearly, the equational theory of paramedial cancellation groupoids is just the least cancellative equational theory containing the paramedial law. We will show that this equational theory is also decidable (Theorem 4.1), that it is a proper extension of the equational theory of paramedial groupoids (Theorem 4.3), and that whenever two terms are unrelated with respect to this equational theory, then their squares are also unrelated (Theorem 4.7). The results can be compared with those of [2] and [3] for medial groupoids.

Term Rewrite Systems for Lattice Theory

It is shown that, even though there is a very well-behaved, natural normal form for lattice theory, there is no finite, convergent AC term rewrite system for the equational theory of all lattices.

Finitely generated algebraic structures with various divisibility conditions

Abstract. Infinite fields are not finitely generated rings. A similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutative semiring that is a group with respect to multiplication can be two-generated only if it belongs to the subclass of additively idempotent semirings; this class is equivalent to

Subdirectly irreducible semilattices with an automorphism

\ell

Definability in substructure orderings, IV: Finite lattices

-groups.

Definability in Substructure Orderings, II: Finite Ordered Sets

In other words, SA is the variety of semilattices with one automorphism (which is, as well as its inverse, considered as an additional fundamental operation). The aim of this paper is to find all subdirectly irreducible algebras in SA . A universal algebra A is said to be subdirectly irreducible (shortly, an SI algebra) if it contains more than one element and among all the nontrivial congruences of A there exists a least one; nontrivial means different from idA = {(a, a); a ∈ A} . The largest example of an SI algebra in SA is the algebra P(Z) defined as follows. Its underlying set is the set of all subsets of Z (where Z denotes the set of integers); the operations are defined by

Selfdistributive groupoids of small orders

Let

Simple zeropotent paramedial groupoids are balanced

{\mathcal{L}}