Towards Minimal Binding Varieties of Lattices

By classical results of Ambrust and Schmidt, every monoid is isomorphic to the endomorphism monoid of an algebra. Thus if we want to investigate in which varieties (or generally, in a concrete category) each monoid is isomorphic to the endomorphism monoid of an algebra in this variety (or an object of this category) then the important role is played by the concept of a binding category (see the definition below). Moreover, as shows the Hedrlín-Kučera Theorem [13] if the set-theoretical assumption (M) holds (i.e., there is an infinite cardinal α such that each ultrafilter closed under α intersections is trivial) every concrete category can be fully embedded into any binding category. Other properties of binding categories are in [7, 11, 13]. A list of the most important binding categories and the properties of binding categories can be found in an excellent monograph [13].