One-variable equational compactness in partially distributive semilattices with pseudocomplementation

Bulman-Fleming, Sydney; Fleischer, Isidore

doi:10.1090/S0002-9939-1980-0572290-3

One-variable equational compactness in partially distributive semilattices with pseudocomplementation
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by Sydney Bulman-Fleming and Isidore Fleischer

Proc. Amer. Math. Soc. 79 (1980), 505-511

DOI: https://doi.org/10.1090/S0002-9939-1980-0572290-3

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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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