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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On bounded po-semigroups

Author: Zahava Shmuely
Journal: Proc. Amer. Math. Soc. 58 (1976), 37-43
MSC: Primary 06A50
MathSciNet review: 0457316
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Abstract: The bounded po-semigroup $ S$ is investigated by studying its increasing elements $ u( \leq {u^2})$ and decreasing elements $ v( \geq {v^2})$. In particular, in $ S,01( = {0^n}{1^m}),10( = {1^n}{0^m}),010$ and $ 101$ are all idempotents and $ 010 = 01{ \wedge _E}10,101 = 10{ \vee _E}01,E$ the set of idempotents of $ S$ ordered as a subset of $ S$. In $ S,0a1 = 01$ and $ 1a0 = 10$ holds for each $ a \in S$. Consequently, $ S$ has a zero element $ z$ iff $ 01 = 10$ and in that case $ z = 01.S$ cannot be cancellative unless it is trivial. $ {J_0} = S10S \subseteq S$ is the kernel of $ S$ and consists of all (idempotents) $ a \in S$ satisfying $ aSa = a$. Thus when $ S$ is a (zero) simple bounded po-semigroup then $ aSa = \{ a,z\} $ and either $ {a^2} = a$ or $ {a^2} = z$ for each $ a \in S$. When $ S = {X^X}$, the po-semigroup of isotone maps $ f$ on the bounded poset $ X$, then $ {J_0}$ consists of all constant maps on $ X$, hence $ {J_0} \simeq X$. The following generalization of Tarski's fixed point theorem is obtained: Let $ S$ be a complete (lattice and a) po-semigroup and let $ s \in S$ be given. Then the set $ {E_s}({J_s})$ of all elements $ {x_0} \in E( \in {J_0}{\text{ resp}}{\text{.)}}$ satisfying $ s{x_0} = {x_0}s = {x_0}$ is a nonempty complete lattice when ordered as a subset of $ S$.

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Article copyright: © Copyright 1976 American Mathematical Society

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