MATHEMATICA BOHEMICA, Vol. 120, No. 1, pp. 71-81, 1995

$\Sigma$-isomorphic algebraic structures

Ivan Chajda, Petr Emanovsky

Katedra algebry a geometrie, PrF UP Olomouc, Tomkova 40, 779 00 Olomouc; Katedra matematiky, PedF UP Olomouc, Zizkovo nam. 8, 771 40 Olomouc

Abstract: For an algebraic structure $\A =(A,F,R)$ or type $\t$ and a set $\Sigma$ of open formulas of the first order language $L(\t)$ we introduce the concept of $\Sigma$-closed subsets of $\A$. The set $\C_\Sigma(\A)$ of all $\Sigma$-closed subsets forms a complete lattice. Algebraic structures $\A$, $\B$ of type $\t$ are called $\Sigma$-isomorphic if $\C_\Sigma(\A)\cong \C_\Sigma(\B)$. Examples of such $\Sigma$-closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study $\Sigma$-isomorphic algebraic structures in dependence on the properties of $\Sigma$.

Keywords: algebraic structure, closure system, subalgebra, ideal, $\Sigma$-closed subset, $\Sigma$-isomorphic structures

Classification (MSC91): 08A05, 04A05, 06B10

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