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

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Last modified: May 6, 2009