Bistable orderings on a certain class of semigroups (Q1071792)

A partial ordering \(\leq\) of a semigroup (S,\(\cdot)\) is called bistable if for every a,b,x,y\(\in S\), with \(a\leq b\) we have xay\(\leq xby\). The author proves the following result which characterizes those semigroups on which each partial ordering is a bistable one: Theorem: Let (S,\(\cdot)\) be a semigroup with \(| S| \geq 3\). The following are equivalent: i) If a,b,c\(\in S\) are distinct, then there exists an ordering \(\leq\) on S which is bistable, such that \(a\leq b\leq c\) and if \(d\in S\), with \(c\leq d\), then \(c=d\). ii) (S,\(\cdot)\) is a strong inflation of a rectangular band T. iii) If \(\leq\) is a partial ordering of S, then \(\leq\) is a bistable ordering of (S,\(\cdot)\).