On Inverse Transversals of Ordered Regular Semigroups

We first consider an ordered regular semigroup S in which every element has a biggest inverse and determine necessary and sufficient conditions for the subset S ○ of biggest inverses to be an inverse transversal of S. Such an inverse transversal is necessarily weakly multiplicative. We then investigate principally ordered regular semigroups S with the property that S ○ is an inverse transversal. In such a semigroup we determine precisely when the set S ☆ of biggest pre-inverses is a subsemigroup and show that in this case S ☆ is itself an inverse transversal of a subsemigroup of S. The ordered regular semigroup of 2 × 2 boolean matrices provides an informative illustrative example. The structure of S, when S ☆ is a group, is also described.