Star-linear equational theories of groupoids

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. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

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, T ^ϖ} is definable, whereT ^ϖ 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 research of both authors was supported by National Science Foundation Grant No. DMS-8302295     

Notes on equational theories of relations

We describe explicitly the free algebras in the equational class generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure and neutral elements 0 (empty relation) and 1 (identity relation). We show the corresponding equational theory is decidable by reducing the problem to a question about regular sets. Similar results are given for two related equational theories.Presented by R. W. Quackenbush.Partially supported by a joint grant from the NSF and the Hungarian Academy of Sciences.Partially supported by a grant from the Hungarian National Foundation for Scientific Research and a joint grant from the NSF and the Hungarian Academy of Sciences. ^v-semirings of 1-closed regular sets. On the basis of this characterization, we conjectured that a set of equational axioms for the variety REL^v consists of equational axioms for the variety L^v and the equation (10). Recently, this conjecture has been proved in [6].     

Bounded and well-placed theories in the lattice of equational theories

In Memory of Evelyn Nelson     

Further properties of lattices of equational theories

Presented by W. Taylor.     

Universal theories for partially commutative metabelian groups

We look at properties of partially commutative metabelian groups and of their universal theories. In particular, it is shown that two partially commutative metabelian groups defined by cycles are universally equivalent if and only if the cycles are isomorphic.     

Congruences,equational theories and lattice representations

We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday     

A property of the lattice of equational theories

It had been conjectured that any algebraic lattice having a compact one could be represented as the lattice of equational theories extending some theory. However, we show that each lattice having such a representation satisfies a nontrivial quasidistributivity condition. In particular,M ₃ has no such representation.To the memory of András HuhnPresented by Walter Taylor.     

On equational theories of varieties of anticommutative rings

The existence of a finitely based variety of anticommutative rings (in the sense of the identityx ²=0) with unsolvable equational theory is proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 230–245, February, 1999.     

Structural diversity in the lattice of equational theories

This paper is principally concerned with conditions under which various partition lattices are isomorphic to intervals in either the lattice of equational theories extending a given equational theory or the lattice of subtheories of a given equational theory. This paper is for Elizabeth Eldridge. Presented by W. Taylor.