Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.Comment: Presented on International conference "Order, Algebra and Logics", Vanderbilt University, 12-16 June, 2007 25 pages, 2 figure

LATTICES OF QUASI-EQUATIONAL THEORIES AS CONGRUENCE LATTICES OF SEMILATTICES WITH OPERATORS

Abstract. We show that for every quasivariety K of relational structures there is a semilattice S with operators such that the lattice of quasiequational theories of K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to Con(S,+,0, F). It is known that if S is a join semilattice with 0, then there is a quasivariety Q such that the lattice of theories of Q is isomorphic to Con(S,+,0) (with no operators). We prove that if S is a semilattice having both 0 and 1 with a group G of operators acting on S, then there is a quasivariety W such that the lattice of theories of W is isomorphic to Con(S,+,0, G). 1. Motivation an

Finite atomistic lattices that can be represented as lattices of quasivarieties

We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11]

REFLECTIONS ON LOWER BOUNDED LATTICES

Abstract. Lattices in the variety LB(k), lower bounded lattices of rank k, are characterized. A sufficient condition for a lattice to be lower bounded is given, and used to produce a new example of a non-finitely-generated lower bounded lattice. Lattices that are subdirect products of finite lower bounded lattices are characterized. 1

Largest Extension of a Finite Convex Geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry