FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 5, PAGES 85-92

On quasiorder lattices and topology lattices of algebras

A. V. Kartashova

Abstract

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In this paper, it is shown that the dual $ \mathop {\widetilde {\mathrm {Qord}}}\mathfrak A $ of the quasiorder lattice of any algebra $ \mathfrak A $ is isomorphic to a sublattice of the topology lattice $ \Im (\mathfrak A) $. Further, if $ \mathfrak A $ is a finite algebra, then $ \mathop {\widetilde {\mathrm {Qord}}}\mathfrak A \cong \Im (\mathfrak A) $. We give a sufficient condition for the lattices $ \mathop {\widetilde {\mathrm {Con}}}\mathfrak A $, $ \mathop {\widetilde {\mathrm {Qord}}}\mathfrak A $, and $ \Im (\mathfrak A) $ to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.

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