Geodesic
Left orders in regular H-Semigroups II
Glasgow mathematical journal, Tome 32 (1990) no. 1, pp. 95-108

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We make the convention that if a is an element of a semigroup Q then by writing a–1 it is implicit that a lies in a subgroup of Q and has inverse a–1 in this subgroup; equivalently, a H a2 and a–1 is the inverse of a in Ha.A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as a–1b where a, b ∈ S and, in addition, every element of S satisfying a weak cancellation condition which we call square-cancellable lies in a subgroup of Q. The notions of right order and semigroup of right quotients are defined dually; if S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S. 
Gould, Victoria. Left orders in regular H-Semigroups II. Glasgow mathematical journal, Tome 32 (1990) no. 1, pp. 95-108. doi: 10.1017/S0017089500009101
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