On bounded po-semigroups

Author:
Zahava Shmuely

Journal:
Proc. Amer. Math. Soc. **58** (1976), 37-43

MSC:
Primary 06A50

DOI:
https://doi.org/10.1090/S0002-9939-1976-0457316-9

MathSciNet review:
0457316

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Abstract: The bounded po-semigroup is investigated by studying its increasing elements and decreasing elements . In particular, in and are all idempotents and the set of idempotents of ordered as a subset of . In and holds for each . Consequently, has a zero element iff and in that case cannot be cancellative unless it is trivial. is the kernel of and consists of all (idempotents) satisfying . Thus when is a (zero) simple bounded po-semigroup then and either or for each . When , the po-semigroup of isotone maps on the bounded poset , then consists of all constant maps on , hence . The following generalization of Tarski's fixed point theorem is obtained: Let be a complete (lattice and a) po-semigroup and let be given. Then the set of all elements satisfying is a nonempty complete lattice when ordered as a subset of .

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0457316-9

Article copyright:
© Copyright 1976
American Mathematical Society