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On the subalgebra lattice of unary algebras. (English) Zbl 0988.08004
The article is devoted to the problem of isomorphism of unary partial algebras.
D. Sachs [Can. J. Math. 14, 451-460 (1962; Zbl 0105.25204)] has shown that two Boolean algebras are isomorphic if and only if their subalgebra lattices are isomorphic. In the article generalizations for the case of unary partial algebras are obtained. Necessary and sufficient conditions are given for two arbitrary unary partial algebras to have isomorphic strong subalgebra lattices. Solutions of some problems for unary algebras and their lattices of subalgebras are also obtained.
In the article a graph language offered by K. Pióro [“On some non-obvious connections between graphs and unary partial algebras”, Czech. Math. J. 50, 295-320 (2000)] is used. The concept of quotient algebra and concepts of contraction of a set of vertices in a graph to digraphs and equivalence relations are generalized. For a digraph \(G\) we can construct various digraphs with strong subdigraph lattices isomorphic to \(S_s(G)\). For unary partial algebras, new unary partial algebras with the same (up to isomorphism) strong subalgebra lattice are constructed.

08A55 Partial algebras
08A60 Unary algebras
08A30 Subalgebras, congruence relations
Zbl 0105.25204
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