In this paper we study big convexity theories, that is convexity theories that are not necessarily bounded. As in the bounded case (see [4]) such a convexity theory Γ gives rise to the category ΓC of (left) Γ-convex modules. This is an equationally presentable category, and we prove that it is indeed an algebraic category over Set. We also introduce the category ΓAlg of Γ-convex algebras and show that the category Frm of frames is isomorphic to the category of associative, commutative, idempotent DU-convex algebras satisfying additional conditions, where D is the two-element semiring that is not a ring. Finally a classification of the convexity theories over D and a description of the categories of their convex modules is given.
@article{urn:eudml:doc:41260,
title = {Convexity theories 0 fin. Foundations.},
journal = {Publicacions Matem\`atiques},
volume = {40},
year = {1996},
pages = {469-496},
mrnumber = {MR1425632},
zbl = {0870.18005},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41260}
}
Kleisli, Heinrich; Röhrl, Helmut. Convexity theories 0 fin. Foundations.. Publicacions Matemàtiques, Tome 40 (1996) pp. 469-496. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41260/