A class of infinite convex geometries

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.Comment: 10 page

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

CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Comment: 14 pages, 4 figure

Sublattices of lattices of order-convex sets, I. The main representation theorem

For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some lattice of the form Co(P) iff L satisfies (S), (U), and (B). Furthermore, if L has an embedding into some Co(P), then it has such an embedding that preserves the existing bounds. If L is finite, then one can take P finite, of cardinality at most $2n^2-5n+4$, where n is the number of join-irreducible elements of L. On the other hand, the partially ordered set P can be chosen in such a way that there are no infinite bounded chains in P and the undirected graph of the predecessor relation of P is a tree

Subprojective lattices and projective geometry

AbstractThe class of lattices we are interested in (subprojective lattices), can be gotten by taking the MacNeille completions of the class of complemented, modular, atomic lattices. McLaughlin showed that subprojective lattices can be represented as the lattices of W-closed subspaces of a vector space U in duality with a vector space W. In this paper, we give a characterization of subprojective lattices in terms of atoms and dual atoms, by means of an incidence space satisfying self-dual axioms. In the finite-dimensional case, a subprojective lattice is projective, and hence our self-dual axioms characterize finite-dimensional projective spaces in terms of points and hyperplanes. No numerical parameters appear explicitly in these axioms. For each subprojective lattice L with at least three elements, we define a projective envelope P(L) for it. P(L) is a projective lattice and there is a natural inf-preserving injection of L into P(L). This injection has other important properties which we take as the definition of a geometric map. In the course of studying geometric maps, we obtain a lattice theoretic proof of Mackey's result that the join of a U-closed subspace of V and a finite-dimensional subspace is U-closed, where (U, V) form a dual pair of vector spaces over a division ring. Furthermore, we show that if L is a subprojective lattice, P a projective lattice, and ψ: L → P a geometric map, then P is isomorphic to the projective envelope P(L) of L. The paper presents many other properties of subprojective lattices. It concludes with a characterization of subprojective lattices which are also projective

A class of infinite convex geometries

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are give