Monadic Bounded Algebras

<p>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.</p>

Monadic Bounded Algebras

<p>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.</p>

A Theory of Congruences and Birkhoff’s Theorem for Matroids

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

Varieties of ∗-regular rings

Given a subdirectly irreducible ∗-regular ring R, we show that R is a homomorphic image of a regular ∗-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular. For any subdirect product of artinian ∗-regular rings we construct a unit-regular and ∗-clean extension within its variety.

Subgroups of direct products of groups invariant under the action of permutations on factors

We study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

Equality Logic

In this paper, we introduce and study a corresponding logic toequality-algebras and obtain some basic properties of this logic. We provethe soundness and completeness of this logic based on equality-algebrasand local deduction theorem. Then we introduce the concept of (prelinear)equality-algebras and investigate some related properties. Also, westudy -deductive systems of equality-algebras. In particular, we provethat every prelinear equality-algebra is a subdirect product of linearly orderedequality-algebras. Finally, we construct prelinear equality logicand prove the soundness and strong completeness of this logic respect toprelinear equality-algebras.

Subgroups of 3-Factor Direct Products

Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.