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closure system; isomorphism; lattice of $\Sigma$-closed subsets; subalgebras; ideals; algebraic structure; $\Sigma$-closed subset; $\Sigma$-isomorphic structures
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 $\Sigma$.
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