Completions for partially ordered semigroups (Q1821128)

A standard completion Y is a function which assigns to each poset P a system YP of lower ends of P such that (i) YP contains all principal lower ends of P and (ii) YP is closed under arbitrary set intersection. A detailed study of such completions in its natural categorical setting has been published recently by the first-named author [Quaest. Math. 9, 149- 206 (1986; Zbl 0602.06002)]. The present paper extends and adjusts these results to the setting of partially ordered semigroups. The key notion is that of a Y-semigroup: S is such iff S is a po-semigroup, Y is a standard completion and all left and right translations of S are Y-continuous, i.e., for every \(y\in S\), \(V\in YS\) we have \(\{\) \(x\in S\); \(y\cdot x\in V\}\in YS\) and \(\{\) \(x\in S\); \(x\cdot y\in V\}\in YS\). For any Y-semigroup S the completion YS is shown to be a complete residuated semigroup, containing an isomorphic copy of S as subsemigroup under the canonical principal ideal embedding. Under rather weak assumptions on Y (Y is compositive, i.e., semigroup homomorphisms \(f: S\to S'\) between Y-semigroups S, S' are already Y- continuous provided inverse images of principal ideals in S' are in YS), the category of complete residuated semigroups is a reflective subcategory of the category of Y-semigroups. The final section of the paper embarks on a careful study of standard extensions of posets which are only conditionally complete. The right setting here is that of conditionable completions Y: Y is such if for each poset P, the natural inclusion \(Y^ 0P\to YP\) is Y-continuous, where \(Y^ 0P\) consists of all nonempty upper bounded members of YP. The results obtained are then extended to the case of Y-semigroups; however, \(Y^ 0S\) need not be a residuated semigroup any longer. This well-written paper unifies and provides a uniform background for many results scattered through the literature.