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Organizers |
Convex Geometries: recent development
by
K. Adaricheva
Institute of Mathematics of SB RAS, Novosibirsk
Convex geometries are defined in combinatorics as the finite closure systems with the anti-exchange axiom. Via the lattices of closed sets they can be linked to the lattices with the unique irredundant decompositions that were studied in 40s by R.Dilworth.
In recent paper by K.Adaricheva, V.Gorbunov and V.Tumanov "Join-semidistributive lattices and convex geometries'' (to appear in Adv.Math.) we discover a close connection
of convex geometries with lattices satisfying the quasi-identity of join-semidistributivity:
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In particular, the lattice of closed sets of any finite convex geometry is join-semidistributive, and every finite join-semidistributive lattice can be embedded into the lattice of convex sets of some convex geometry. This also determines the place of the class of join-semidistributive lattices in the whole lattice hierarchy as a class that in some sense opposes to the class of modular lattices, the latter often being linked to the closure systems with the exchange-axiom.
The paper above introduces the general notion of a convex geometry as a (not necessarily finite) closure system with the anti-exchange axiom. This allows studying a wide class of closure systems that appear in different mathematical disciplines.
In the talk we will overview the recent results about convex geometries (including the author's work but not limited to it). We mention the progress that was done toward the solutions to some problems raised in the paper cited above and tell about recent studies of such key examples of convex geometries as lattices of convex subsets of partially ordered sets, lattices of suborders of partial orders, lattices of algebraic subsets of algebraic lattices and lattices of convex subsets of vector spaces.
Date received: January 31, 2003
Copyright © 2003 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 # cake-48.