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Semigroups of Left Quotients—The Layered Approach

Abstract 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 expressed as a ♯ b where a, b ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a ♯ denotes the inverse of a in a subgroup of Q. We say that a left order S is straight in Q if in the above definition we can insist that a ℛ b in Q. A complete characterisation of straight left orders in terms of embeddable ∗-pairs is available. In this article we adopt a different approach, based on partial order decompositions of semigroups. Such decompositions include semilattice decompositions and decompositions of a semigroup into principal factors or principal ∗-factors. We determine when a semigroup that can be decomposed into straight left orders is itself a straight left order. This technique gives a unified approach to obtaining many of the early results on characterisations of straight left orders.