The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Sums and extensions of vector lattice homomorphisms

This paper gives a summary account, with minimal or no proofs, first of some results which characterize order bounded linear operators which are sums of lattice homomorphisms, or more generally of orthomorphisms; and secondly of theorems concerning extensions of vector lattice homomorphisms (theorems of Hahn-Banach type if you will). In all cases we assume that domain and range are vector lattices and that the range is Dedekind complete. The results vary from historical (pre-1940) to recent (1990). The most recent work, on sums of lattice homomorphisms, is covered in §1 and the more classical work on extension theorems is dealt with in §2.     

On sublattice lattice varieties

Presented by R. W. Quackenbush.     

The lattice of semigroup varieties

Semigroups are considered here in terms of the identities which they may satisfy with the emphasis on the identities and on classes of semigroups defined by identities, rather than on the semigroups themselves. These classes, called varieties, form a lattice under inclusion and it is illuminating to interpret properties of semigroup identities and varieties in terms of this lattice, its cardinality, its atoms, its infinite chains, and so on. This paper is an expansion of addresses to the Southeastern Section of the Mathematical Association of America at East Carolina University, March 1968 and to the Symposium on Semigroups and Rings at the University of Puerto Rico, Mayaguez, March 1970. The preparation of the paper was supported in part by NSF Grants GP 6597 and GP 20638.     

Lattice of subalgebras in the finitely generated varieties of MV-algebras

In this paper, we use the theory of natural duality to study subalgebra lattices in the finitely generated varieties of MV-algebras. With this tool, we obtain the dual atomicity of these lattices, and characterize the members of these varieties in which every subalgebra is an intersection of maximal subalgebras. Then, we determine the algebras that have a modular or distributive lattice of subalgebras.     

Two finitely generated varieties having no infinite simple members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote's negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

     

Bounding essential arities of term operations in finitely decidable varieties

Let A be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical \({\tau}\). We provide an improved bound on the number of variables in which a term can be sensitive to changes within \({\tau}\). We utilize a multisorted algebraic construction, amalgamating the methods developed by Valeriote and McKenzie for the investigation of strongly abelian locally finite decidable varieties with those of Idziak for locally finite congruence modular finitely decidable varieties.     

Nondefinability of projectivity in lattice varieties

The relationship of projectivity between two quotients in a lattice is shown not to be first-order definable, in any nondistributive lattice variety. The proof depends on a special kind of subdirect power construction that shows the existence of arbitrarily long non-shortenable projectivities in such a variety. A similar result holds for weak projectivities. Even so, weak projectivities of bounded length do suffice to determine principal congruences in any variety generated by a finite lattice.