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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512047-4

MathSciNet review:
512047

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Abstract: We consider universal algebras in which *E* is a set of endomorphisms of the semilattice . 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 and every up-directed family in *S* the equality holds, (iv) for each holds for every family in *S*, and (v) for each holds for every up-directed family 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.

**[1]**S. Bulman-Fleming,*On equationally compact semilattices*, Algebra Universalis**2**(1972), 146-151. MR**0308003 (46:7118)****[2]**-,*A note on equationally compact algebras*, Algebra Universalis**4**(1974), 41-43. MR**0360408 (50:12858)****[3]**S. Bulman-Fleming and I. Fleischer,*One-variable equational compactness in partially distributive semilattices with pseudocomplementation*, manuscript.**[4]**G. Grätzer,*Universal algebra*, Van Nostrand, Princeton, N.J., 1968. MR**0248066 (40:1320)****[5]**G. Grätzer and H. Lakser,*Equationally compact semilattices*, Colloq. Math.**20**(1969), 27-30. MR**0238753 (39:117)****[6]**J. Mycielski,*Some compactifications of general algebras*, Colloq. Math.**13**(1964), 1-9. MR**0228405 (37:3986)****[7]**W. Taylor,*Review of several papers on equational compactness*, J. Symbolic Logic**40**(1975), 88-92.**[8]**G. H. Wenzel,*Equational compactness in universal algebras*, Habilitationsschrift, Mannheim, 1971.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512047-4

Keywords:
Semilattice,
equationally compact,
topological semilattice

Article copyright:
© Copyright 1979
American Mathematical Society