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The subalgebra lattice of a finite algebra. (English) Zbl 1292.08003
Summary: The aim of this paper is to characterize pairs \((L,A)\), where \(L\) is a finite lattice and \(A\) a finite algebra, such that the subalgebra lattice of \(A\) is isomorphic to \(L\). Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.

MSC:
08A30 Subalgebras, congruence relations
05C65 Hypergraphs
05C90 Applications of graph theory
06B05 Structure theory of lattices
06B15 Representation theory of lattices
06D05 Structure and representation theory of distributive lattices
08A55 Partial algebras
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