Niefield, Susan B.; Rosenthal, Kimmo I. Constructing locales from quantales. (English) Zbl 0658.06007 Math. Proc. Camb. Philos. Soc. 104, No. 2, 215-234 (1988). A quantale is a complete lattice equipped with an associative binary operation which distributes over arbitrary joins (on either side). A frame (which the authors of this paper call a locale) is a quantale in which the binary operation coincides with meet; equivalently, it is a quantale in which the binary operation is commutative and idempotent and has the top element of the lattice as a unit. In this paper the authors’ main objective is to give an explicit description of the left adjoint to the inclusion functor from frames to quantales; they do this by breaking it down into several steps, forcing the additional properties required of the binary operation one by one. In addition, they establish various results about the lattice of quantic nuclei on an arbitrary quantale Q (which correspond to quotients of Q in the category of quantales). Some applications are given to ideal lattices of rings. Reviewer: P.T.Johnstone Cited in 5 ReviewsCited in 37 Documents MSC: 06F05 Ordered semigroups and monoids 08A30 Subalgebras, congruence relations 16Nxx Radicals and radical properties of associative rings 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) Keywords:left adjoint to the inclusion functor from frames to quantales; lattice of quantic nuclei; ideal lattices of rings PDF BibTeX XML Cite \textit{S. B. Niefield} and \textit{K. I. Rosenthal}, Math. Proc. Camb. Philos. Soc. 104, No. 2, 215--234 (1988; Zbl 0658.06007) Full Text: DOI References: [1] Dixmier, C*-Algebras (1977) [2] DOI: 10.1007/BF00403411 · Zbl 0595.18003 [3] Borceux, Algebra in a Localic Topos with Applications to Ring Theory 1038 (1983) · Zbl 0522.18001 [4] DOI: 10.1112/plms/s3-50.3.385 · Zbl 0569.16003 [5] Simmons, Logic Colloquium 77 pp 239– (1978) [6] DOI: 10.1112/plms/s3-16.1.275 · Zbl 0136.43405 [7] DOI: 10.1016/0166-8641(87)90046-0 · Zbl 0621.06007 [8] Mulvey, Rend. Circ. Mat. Palermo 12 pp 99– (1986) [9] DOI: 10.1112/jlms/s1-44.1.283 · Zbl 0159.33502 [10] Joyal, An Extension of the Galois Theory of Grothendieck (1984) · Zbl 0541.18002 [11] Johnstone, Stone Spaces (1982) [12] Niefield, Proceedings of Category Theory Meeting, Louvain-la-Neuve (1987) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.