MATHEMATICA BOHEMICA, Vol. 120, No. 2, pp. 209-217, 1995

Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure

Ivan Chajda, Petr Emanovsky

Ivan Chajda, katedra algebry a geometrie, Prir. fak. UP Olomouc, Tomkova 38, 779 00 Olomouc; Petr Emanovsky, katedra matematiky, Ped. fak. UP Olomouc, Zizkovo nam. 5, 771 40 Olomouc

Abstract: Let $\Cal A =(A,F,R)$ be an algebraic structure of type $\tau$ and $\Sigma$ a set of open formulas of the first order language $L(\tau)$. The set $C_\Sigma(\Cal A)$ of all subsets of $A$ closed under $\Sigma$ forms the so called lattice of $\Sigma$-closed subsets of $\Cal A$. We prove various sufficient conditions under which the lattice $C_\Sigma(\Cal A)$ is modular or distributive.

Keywords: algebraic structure, closure system, $\Sigma$-closed subset, modular lattice, distributive lattice, convex subset

Classification (MSC91): 08A05, 04A05

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