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**Constructing locales from quantales.**
*(English)*
Zbl 0658.06007

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

### 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
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\textit{S. B. Niefield} and \textit{K. I. Rosenthal}, Math. Proc. Camb. Philos. Soc. 104, No. 2, 215--234 (1988; Zbl 0658.06007)

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