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

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

Full text of the article:

• Compressed Postscript file (225857 Bytes)