Geodesic
Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure
Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 209-217.

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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.
DOI : 10.21136/MB.1995.126220
Classification : 03C05, 04A05, 06A23, 06B10, 06C05, 06D05, 08A05
Keywords: closure system; distributive lattice; lattices of $\Sigma-closed subsets; modular lattice; algebraic structures; $\Sigma$-closed subset; convex subset 
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Chajda, Ivan; Emanovský, Petr. Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure. Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 209-217. doi : 10.21136/MB.1995.126220. http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126220/

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