  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
and b are elements of 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 is related to b by Green's relation R in Q. A
complete characterisation of straight left orders in terms of embeddable
*-pairs is available. In this paper 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.
