The semilattices with distinguished endomorphisms which are equationally compact
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- by Sydney Bulman-Fleming, Isidore Fleischer and Klaus Keimel
- Proc. Amer. Math. Soc. 73 (1979), 7-10
- DOI: https://doi.org/10.1090/S0002-9939-1979-0512047-4
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Abstract:
We consider universal algebras $(S;\{ \wedge \} \cup E)$ in which E is a set of endomorphisms of the semilattice $(S; \wedge )$. It is proved in this paper that such an algebra is equationally compact iff (i) every nonempty subset of S has an infimum, (ii) every up-directed subset of S has a supremum, (iii) for every $s \in S$ and every up-directed family $({d_i})$ in S the equality $s \wedge \vee {d_i} = \vee s \wedge {d_i}$ holds, (iv) for each $f \in E,f( \wedge {s_i}) = \wedge f({s_i})$ holds for every family $({s_i})$ in S, and (v) for each $f \in E,f( \vee {d_i}) = \vee f({d_i})$ holds for every up-directed family $({d_i})$ in S. In addition, it is shown that every equationally compact algebra of this type is a retract (algebraic) of a compact, Hausdorff, 0-dimensional topological one. These results reduce to known ones for semilattices without additional structure.References
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Bibliographic Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 73 (1979), 7-10
- MSC: Primary 08A45
- DOI: https://doi.org/10.1090/S0002-9939-1979-0512047-4
- MathSciNet review: 512047