On left \(\pi\)-regular po-semigroups (Q2744275)

A partially ordered (p.o.) semigroup \((S,\cdot,\leq)\) is called left \(\pi\)-regular if for every element \(a\in S\) there exist \(m>0\), \(x\in S\) such that \(a^m\leq xa^{2m}\). Defining a left ideal \(I\) of \((S,\cdot,\leq)\) as a subset of \(S\) which is at the same time a left ideal of \((S,\cdot)\) and an order-ideal of \((S,\leq)\), three characterizations of such p.o. semigroups in terms of (principal) left ideals are given. In particular, it is shown that a p.o. semigroup \((S,\cdot,\leq)\) is left \(\pi\)-regular iff every left ideal \(I\) of \(S\) has the property that for any \(a\in S\) there exists \(n>0\) such that \(a^{2n}\in I\) implies that \(a^n\in I\). Considering for ``\(\leq\)'' the identity relation this result generalizes the fact that a semigroup \((S,\cdot)\) is left regular (i.e., for every \(a\in S\) there exists \(x\in S\) with \(a= xa^2\)) iff every left ideal \(I\) of \(S\) is semiprime (i.e., if \(a^2\in I\), \(a\in S\), then \(a\in I\)).