On modular lattices generated by chains

We describe the free modular lattice generated by two chains and a single point, under the assumption that there are few meets. Received February 11, 2005; accepted in final form August 11, 2005.     

The variety generated by planar modular lattices

We investigate the variety generated by the class of planar modular lattices. The main result is a structure theorem describing the subdirectly irreducible members of this variety.     

On distributive sublattices of modular lattices

Presented by R. Freese.     

On a class of modular lattices

Translated from Algebra i Logika, Vol. 30, No. 1, pp. 3–14, January-February, 1991.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

On finitely generated lattices of width four

We define an infinite class ?₄ of infinite lattices with the property that every finitely generated infinite lattice of width four contains (up to duality) a sublattice isomorphic to the herringbone or to a member of ?₄. A consequence is that every finitely generated infinite lattice of width four generates a variety of infinite height (in the lattice of varieties of lattices).     

Matching in modular lattices

Let L be a finite lattice. A map f of the join irreducible elements of L to the meet irreducible elements of L is called a matching of L if f is one-to-one and x?f(x) for each join irreducible x. We investigate this conjecture: every finite modular lattice has a matching. The conjecture is verified for certain classes of modular lattices.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.