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Convexity theories 0 fin. foundations

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  • Mathematics


In this paper we study big convexity theories, that is convexity theories that are not necessarily bounded. As in the bounded case (see \cite{4}) such a convexity theory $\Gamma$ gives rise to the category $\Gamma{\Cal C}$ of (left) $\Gamma$-convex modules. This is an equationally presentable category, and we prove that it is indeed an algebraic category over ${\Cal S}et$. We also introduce the category $\Gamma{\Cal A}lg$ of $\Gamma$-convex algebras and show that the category ${\Cal F}rm$ of frames is isomorphic to the category of associative, commutative, idempotent $\Bbb D^U$-convex algebras satisfying additional conditions, where $\Bbb D$ is the two-element semiring that is not a ring. Finally a classification of the convexity theories over $\Bbb D$ and a description of the categories of their convex modules is given.

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