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# Entailment systems for stably locally compact locales

Vickers, Steven (2004) Entailment systems for stably locally compact locales. Theoretical Computer Science, 316 (1-3). pp. 259-296. ISSN 0304-3975

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URL of Published Version: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1G-4BSWJ2J-

Identification Number/DOI: doi:10.1016/j.tcs.2004.01.033

The category SCFrU of stably continuous frames and preframe ho-momorphisms (preserving ¯nite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose
morphisms X ! Y are upper closed relations between the ¯nite powersets FX and FY . Composition of these morphisms is the \cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent
calculus. Thus stably locally compact locales are represented by \entailment systems" (X; ) in which , a generalization of entailment relations,is idempotent for cut composition.
Some constructions on stably locally compact locales are represented
in terms of entailment systems: products, duality and powerlocales.
Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A t B is isomorphic to e A ­ B where e A is the Hofmann-Lawson dual.
For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X.

Type of Work: Article 28 May 2004 (Publication) Schools (1998 to 2008) > School of Computer Science Computer Science Q Science (General)QA75 Electronic computers. Computer scienceQA Mathematics University of Birmingham Elsevier 187 YES

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