Title:

Semigroups of Iquotients

Let Q be an inverse semigroup. A subsemigroup S of Q is a left Iorder in Q or Q is a semigroup of left Iquotients of S, if every element in Q can be written as a—l b where a, b E S and a' is the inverse of a in the sense of inverse semigroup theory. If we insist on a and b being 7Zrelated in Q, then we say that S is straight in Q and Q is a semigroup of straight left Iquotients of S. We give a theorem which determines when two semigroups of straight left Iquotients of given semigroup are isomorphic. Clifford has shown that, to any right cancellative monoid with the (LC) condition, we can associate an inverse hull. By saying that a semigroup S has the (LC) condition we mean for any a, b E S there is an element c E S such that SanSb = Sc. According to our notion, we can regard such a monoid as a left Iorder in its inverse hull. We extend this result to the left ample case where we show that, if a left ample semigroup has the (LC) condition, then it is a left Iorder in its inverse hull. The structure of semigroups which are semilattices of bisimple inverse monoids, in which the set of identity elements forms a subsemigroup, has been given by Cantos. We prove that such semigroups are strong semilattices of bisimple inverse monoids. Moreover, they are semigroups of left Iquotients of semigroups with the (LC) condition, which are strong semilattices of right cancellative monoids with the (LC) condition. We show that a strong semilattice S of left ample semigroups with (LC) and such that the connecting homomorphisms are (LC)preserving, itself has the (LC) condition and is a left Iorder in a strong semilattice of inverse semigroups. We investigate the properties of left Iorders in primitive inverse semigroups. We give necessary and sufficient conditions for a semigroup to be a left Iorder in a primitive inverse semigroup. We prove that a primitive inverse semigroup of left Iquotients is unique up to isomorphism. We study left Iorders in a special case of a bisimple inverse wsemigroup, namely, the bicyclic monoid. Then, we generalise this to any bisimple inverse wsemigroup. We characterise left Iorders in bisimple inverse wsemigroups.
