Tue, 10 Nov 2020, 1 pm MST
A “form” is a category equipped with abstract subobject posets for its objects and “connections” between those posets induced by morphisms of the category. In this talk we present an application of this structure in developing a self-dual approach to homomorphism theorems for group-like universal algebras. We call a form satisfying the axioms necessary for obtaining the standard isomorphism theorems a “noetherian form”. Examples of noetherian forms abound (in some of the examples, though, the isomorphism theorems trivialize). For instance, any bounded lattice can be viewed as a noetherian form with a trivial base category. Furthermore, not only group-like universal algebras (e.g., modules, groups, rings with or without unit, loops, etc.), but even any variety of universal algebras can be viewed as a noetherian form, thanks to the fact that the category of sets has the structure of a noetherian form (a non-trivial fact, since the two natural forms, that of subsets and that of equivalence relations, are both non-neotherian). If time allows, we will also discuss some combinatorial problems suggested by the notion of a noetherian form.