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Precoherent quantale completions of partially ordered semigroups. (English) Zbl 1444.06011

There exists the concept of coherent locale, and it is well-known that a locale is coherent if and only if it is isomorphic to the locale of ideals of a distributive lattice (cf., e.g., [P. T. Johnstone, Stone spaces. Cambridge: Cambridge University Press (1982; Zbl 0499.54001)]). A non-commutative version of this result states that a quantale is coherent if and only if it is isomorphic to the quantale of all ideals of an \(m\)-semilattice (Proposition 4.2 on page 209 of [J. Paseka, Arch. Math., Brno 22, 203–210 (1986; Zbl 0612.06012)]; see also [K. Keimel, Acta Math. Acad. Sci. Hung. 23, 51–69 (1972; Zbl 0265.06016)]). The present paper studies those completions of partially ordered semigroups, which give precoherent quantales. Recall that a quantale \((Q,\otimes, \leqslant)\) is said to be precoherent provided that \((Q, \leqslant)\) is an algebraic lattice, and \(K(Q)\) (the set of compact elements of \(Q\)) is closed under \(\otimes\). A precoherent quantale \(Q\) is then called coherent provided that the top element \(\top_Q\) of \(Q\) is compact (Definition 2.9 on page 620).
The paper first provides a representation theorem for coherent quantales, namely, shows that a quantale \(Q\) is coherent if and only if \(Q\) is isomorphic to the quantale of all distributive ideals of a partially ordered semigroup with a top element (Theorem 3.9 on page 623). Notice that in case of \(m\)-semilattices, distributive ideals are precisely ideals, which gives back the above-mentioned result for \(m\)-semilattices. The next result shows that precoherent quantale completions of a partially ordered semigroup \(S\) are in one-to-one correspondence with quantic quotients of the quantale \(\mathcal{P}(S)\) (the powerset of \(S\)) with respect to algebraic consistent quantic nuclei on \(\mathcal{P}(S)\) (Corollary 3.16 on page 625). Moreover, Theorem 3.24 on page 627 gives the smallest and the largest precoherent quantale completion of \(S\) (notice that the set of all algebraic consistent quantic nuclei on \(\mathcal{P}(S)\) is a complete lattice and thus has the smallest and the largest element). Additionally, the authors provide the necessary and sufficient conditions on \(S\) for its Frink completion to be a coherent quantale (Corollary 3.33 on page 629). Lastly, motivated by the result that quantales are precisely the injective objects in a certain category of partially ordered semigroups (namely, by Theorem 4 on page 374 of [X. Zhang and V. Laan, Proc. Est. Acad. Sci. 63, No. 4, 372–378 (2014; Zbl 1332.06047)], which relies on the arguments of Theorem 4.1 on page 342 of [J. Lambek et al., Theory Appl. Categ. 26, 338–348 (2012; Zbl 1259.06019)]), the authors define a category of algebraic partially ordered semigroups APoSgr\({}_{\leq}\) and a class of its morphism \(\mathscr{E}_{\leq}\) such that \(\mathscr{E}_{\leq}\)-injective objects in APoSgr\({}_{\leq}\) are precisely precoherent quantales (Theorem 4.4 on page 631). The authors, however, claim to be unable to find \(\mathscr{E}_{\leq}\)-injective hulls in APoSgr\({}_{\leq}\) (Remark 4.7(1) on page 632).
The paper is sufficiently well written (with a number of typos, like, e.g., “procoherent” instead of “precoherent” on page 618), contains nearly all of its required preliminaries, and will be of interest to all the researchers, who work with the theory of ordered algebraic structures.

MSC:

06F07 Quantales
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B10 Lattice ideals, congruence relations
06B23 Complete lattices, completions
06F05 Ordered semigroups and monoids
18B99 Special categories
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References:

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