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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On subalgebra lattices of universal algebras

Author: A. A. Iskander
Journal: Proc. Amer. Math. Soc. 32 (1972), 32-36
MSC: Primary 08A25
MathSciNet review: 0292733
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Abstract: If A is a universal algebra, $ S(A)$ is the lattice of all subalgebras of A. If $ B \subseteq A \times A,{B^\ast}$ is $ \{ (x,y):(y,x) \in B\} $.

Theorem. Let $ {L_1},{L_2},{L_3}$ be algebraic lattices such that $ \vert{L_1}\vert,\vert{L_2}\vert > 1$. Let $ {\alpha _i}$ be an involutive automorphism of $ {L_i},i = 1,2$. Then there are two universal algebras $ {A_1},{A_2}$ of the same similarity type, having the properties:

(a) there are lattice isomorphisms $ {\beta _i}$ of $ {L_i}$ onto $ S({A_i} \times {A_i}),i = 1,2$, and $ {\beta _3}$ of $ {L_3}$ onto $ S({A_1} \times {A_2})$;

(b) $ (l{\alpha _i}){\beta _i} = {(l{\beta _i})^\ast},l \in {L_i},i = 1,2$.

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Keywords: Algebraic lattice, compact elements, universal algebra, subalgebra, involutive automorphism, partial operations, free extensions
Article copyright: © Copyright 1972 American Mathematical Society