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An inverse semigroup is called proper if the equations a e = e = e 2 ae = e = {e^2} together imply a 2 = a {a^2} = a . In a previous paper, with the same title, the author proved that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup. In this paper a structure theorem is given for all proper inverse semigroups in terms of partially ordered sets and groups acting on them by order automorphisms. As a consequence of these two theorems, and Preston’s construction for idempotent separating congruences on inverse semigroups, one can give a structure theorem for all inverse semigroups in terms of groups and partially ordered sets.

A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

A regular semigroup S is locally inverse if each local submonoid eSe, e an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism 0 which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.

A partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism 9 of S onto a partially ordered group such that {x e S: xty < x9) has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well. Partially ordered regular semigroupsIt is well known that the idempotents of a regular semigroup 5 can be partially ordered by setting euf if and only if e = ef = fe. This partial order is called the natural partial order on the idempotents of S. We shall say that a partially ordered semigroup S is naturally partially ordered if the imposed partial order < extends the natural partial order on the idempotents; that is, if e = ef = fe implies e < f. Note there is no assumption in the definition that e < / implies e = ef -fe. As we shall see, the existence of a partial order, extending the natural partial

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