Convexity theories 0 fin. foundations

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

Totally convex algebras

summary:By definition a totally convex algebra $A$ is a totally convex space $|A|$ equipped with an associative multiplication, i.e\. a morphism $\mu :|A|\otimes |A|\longrightarrow |A|$ of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces