M. V. Schwidefsky, “Decompositions in complete lattices III. Unique irredundant decompositions and convex geometries”, Algebra Logika, 2017, Volume 56, Number 5,Pages <nobr>613

Decompositions in complete lattices III. Unique irredundant decompositions and convex geometries

M. V. Schwidefsky^ab

^a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
^b Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090 Russia

Abstract: We give a characterization of complete strongly dually atomic lattices having unique irredundant decompositions which are also canonical. It is shown that all known characterizations of lattices with unique irredundant decompositions are a consequence of this result. In addition, upper continuous closure lattices of convex geometries with (unique) irredundant decompositions are characterized.

Keywords: closure space, convex geometry, irredundant decomposition, join-semidistributive lattice, locally distributive lattice, lower continuous lattice, minimal decomposition, semimodular lattice, strongly atomic lattice, upper continuous lattice, weakly atomic lattice.

UDC: 512.56

Received: 05.04.2016
Revised: 10.11.2016

DOI: 10.17377/alglog.2017.56.506