On bounded po-semigroups

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 $S$.