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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The semilattices with distinguished endomorphisms which are equationally compact


Authors: Sydney Bulman-Fleming, Isidore Fleischer and Klaus Keimel
Journal: Proc. Amer. Math. Soc. 73 (1979), 7-10
MSC: Primary 08A45
MathSciNet review: 512047
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0512047-4
PII: S 0002-9939(1979)0512047-4
Keywords: Semilattice, equationally compact, topological semilattice
Article copyright: © Copyright 1979 American Mathematical Society