Cancellative orders

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Publication details

JournalSemigroup forum
DatePublished - 1997
Issue number2
Number of pages11
Pages (from-to)185-195
Original languageEnglish


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 q is an element of Q can be written as q = a*b for some a, b is an element of S where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the D-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S.

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