کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4662662 1633504 2010 16 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Fundamental results for pointfree convex geometry
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات منطق ریاضی
پیش نمایش صفحه اول مقاله
Fundamental results for pointfree convex geometry
چکیده انگلیسی

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Annals of Pure and Applied Logic - Volume 161, Issue 12, September 2010, Pages 1486-1501