Join-semidistributive lattices and convex geometries

K. V. Adaricheva, V. A. Gorbunov, V. I. Tumanov

Research output: Contribution to journalArticle

50 Citations (Scopus)

Abstract

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice Sp(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, Sp (A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.

Original languageEnglish
Pages (from-to)1-49
Number of pages49
JournalAdvances in Mathematics
Volume173
Issue number1
DOIs
Publication statusPublished - Jan 15 2003

Keywords

  • Anti-exchange property
  • Antimatroid
  • Atomistic
  • Biatomic
  • Convex geometry
  • Join-semidistributive
  • Lattice
  • Quasivariety

ASJC Scopus subject areas

  • Mathematics(all)

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