Monadic Bounded Algebras

The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given.