Semigroups of straight I-quotients: a general approach

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q, and Q is a semigroup of left I-quotients of S, if every element in Q can be written as a^{-1}b, where a, b \in S and a^{-1} is the inverse of a in the sense of inverse semigroup theory. If we insist on a and b being R-related in Q, we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. In Chapter 4, we give two equivalent sets of necessary and sufficient conditions for a semigroup to be a straight left I-order. The first set of conditions is in terms of two binary relations and an associated partial order and the proof relies on the meet structure of the L-classes of inverse semigroups. The second set of conditions in terms of two binary relations and a ternary relation and the proof is purely algebraic. We characterise right ample straight left I-orders that are embedded as a unary semigroup into their semigroups of straight left I-quotients. As a special case of this, we characterise two-sided ample left I-orders that are embedded into their semigroups of left I-quotients as (2,1,1)-algebras. Straight left I-orders always intersect every L-class of their semigroup of straight left I-quotients. We characterise straight left I-orders that intersect every R-class of their semigroup of straight left I-quotients. We use this to prove that if a semigroup S has both a semigroup of straight left I-quotients, Q, and a semigroup of straight right I-quotients, P, then P and Q are isomorphic if and only if their R and L relations restricted to S are equal. We characterise left I-orders whose semigroups of quotients have a chain of idempotents. As a special case of this, we characterise left I-orders in inverse ω-semigroups. We determine when two semigroups of straight left I-quotients are isomorphic.