Compatible orders on the bicyclic semigroup

Abstract

In this paper we determine those compatible partial orders on the bicyclic semigroup B which turn it into a semilatticed semigroup. We shall see that there are exactly four distinct compatible total orderings on B. These are the only compatible orderings which turn B into a lattice ordered semigroup. On a group every compatible semilattice ordering is a lattice ordering. However this is not the case with inverse semigroups. Indeed, the situation regarding semilattice orderings on the bicyclic semigroups is much richer. There are four infinite families of compatible semilattce orderings on B. Two of these families turn B into a V-semilatticed semigroup; two of the families turn it into a ^-semilatticed semigroup.