The Universal Theory of First Order Algebras and Various Reducts

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras (where we restrict attention to the positive existential formulas) and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.Comment: 30 page

Some Case Studies in Algebra Motivated by Abstract Problems of Language

This thesis concerns three different topics. The first has to do with axiomatizing the universal theory of certain classes of multisorted algebras arising from intersection, union, and other first order operations on relations. The second has to do with axiomatizing certain classes of actions arising from intersection and union, and axiomatizing certain classes of posets arising from actions arising from intersection. The third has to do with understanding under what conditions morphisms between two structures can be finitely determined