On a strong property of the weak subalgebra lattice |
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Authors: | K. Pióro |
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Affiliation: | (1) Institute of Mathematics, Warsaw University, ul. Banacha 2, PL-02-097 Warsaw, Poland, e-mail: kpioro@mimuw.edu.pl, PL |
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Abstract: | In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998. |
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