In this article we present several logical schemes. The scheme FraenkelA2 deals with a non empty set A , a binary functor F yielding a set, and two binary predicates P , Q , and states that: {F (s, t);s ranges over elements of A , t ranges over elements of A : P [s, t]} is a subset of A provided the following condition is satisfied: • For every element s of A and for every element t of A holds ...

It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categorie...

By the sorting method of vertices, the equitable chromatic number and the equitable chromatic threshold of the Cartesian products of wheels with bipartite graphs are obtained. Key–Words: Cartesian product, Equitable coloring, Equitable chromatic number, Equitable chromatic threshold

Introduction THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.) Firstly, the cartesian product of identification maps is not, in general, an identification map. As a consequence certain natural products such as the join and smash...