Orders in strict regular semigroups

A subsemigroup S of a semigroup Q is an order in Q if for every q is an ele ment of Q there exist a, b, c, d is an element of S such that q = a(-1) = c d(-1), where a and d are contained in (maximal) subgroups of Q, and a(-1) a nd d(-1) are their inverses in these subgroups. A regular semigroup S is st rict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger's model of a strict regu lar semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q! for extendibil ity to an involution on Q.