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Proceedings of the American Mathematical Society

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

 

 

One-variable equational compactness in partially distributive semilattices with pseudocomplementation


Authors: Sydney Bulman-Fleming and Isidore Fleischer
Journal: Proc. Amer. Math. Soc. 79 (1980), 505-511
MSC: Primary 06A12; Secondary 08A45
DOI: https://doi.org/10.1090/S0002-9939-1980-0572290-3
MathSciNet review: 572290
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Abstract: A universal algebra A is called one-variable equationally compact if every system of equations with constants in A involving a single variable x, every finite subsystem of which has a solution in A, has itself a solution in A. The one-variable equationally compact semilattices with pseudocomplementation $ \langle S; \wedge {,^ \ast },0\rangle $ which satisfy the partial distributive law $ x \wedge {(y \wedge z)^ \ast } = (x \wedge {y^ \ast }) \vee (x \wedge {z^ \ast })$ are characterized, and as a consequence we are able to describe the one-variable compact Stone semilattices. Similar considerations yield a characterization of the one-variable equationally compact Stone algebras, extending a well known result for distributive lattices.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0572290-3
Keywords: Semilattice, pseudocomplement, equationally compact
Article copyright: © Copyright 1980 American Mathematical Society