Orders in strict regular semigroups (Q1568799)

A subsemigroup \(S\) of a semigroup \(Q\) is an order in \(Q\) if for every \(q\in Q\) there are \(a,b,c,d\in S\) such that \(q=a^{-1}b=cd^{-1}\), where \(a\) and \(d\) are contained in maximal subgroups of \(Q\) and \(a^{-1}\) and \(d^{-1}\) are group inverses; \(Q\) is then called a semigroup of quotients of \(S\). This definition originates from a successful concept in ring theory and takes into account the usual lack of an identity in a semigroup. Orders in semigroups have been described for various classes of semigroups, e.g. completely \(0\)-simple semigroups. In this paper, the author finds all orders in a strict regular semigroup (which can be built up from its completely \(0\)-simple principal factors in a very transparent way). In addition, all involutions on such a semigroup are constructed and it is investigated whether an involution on an order can be extended to its semigroup of quotients.