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Organizers |
Lattices of algebraic subsets
by
Kira Adaricheva
Institue of Mathematics, SB RAS, Novosibirsk, Russia
Lattices of algebraic subsets of complete lattices arise naturally in the most general setting of continuous closure operators.
In the theory of quasivarieties they provide the model for the lattices of subquasivarieties, or Q-lattices for short. The problem of characterization of Q-lattices was raised independently by G. Birkhoff and A.I. Mal'cev. We give an overview of major results describing Q-lattices and further perspectives in that direction.
The lattices of algebraic subsets also provide a key example of a convex geometry, the notion that extends the combinatorial notion of (finite) convex geometry. In general lattice theory it places these lattices into the juxtaposition to the lattices of equivalence relations. If the latter play significant role in classification of lattices according to modular law, lattices of algebraic subsets do that with respect to join-semidistributive law. Both laws are important and somewhere ''opposite'' generalizations of distributive law: they reflect the nature of closure operators with exchange and anti-exchange property, correspondingly.
In this talk we will mention recent development in the lattice theory of convex geometries and also point to some open problems.
Date received: January 31, 2002
Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-64.