About quasivarieties of p-algebras and Wajsberg algebras

A counterexample is given to show that not all quasivarieties of p-algebras lie between two consecutive varieties. It is shown that the quasivariety of p-algebras generated by the finite subdirectly irreducible p-algebras is the entire variety of p-algebras. Also, it is shown that this variety is not structurally complete and the class of its finitely subdirectly irreducible members coincides with the class of its subdirectly irreducible ones. This later result is used to show that there are no strict relatively congruence distributive quasivarieties of p-algebras. Relatively congruence distributive quasivarieties of Wajsberg algebras are characterized. The relative congruence extension property in the classes of p-algebras and Wajsberg algebras is studied. It is proved that in the first class only quasivarieties which are varieties possess this property. In the second one, it is shown that a quasivariety which is relatively congruence distributive or generates a proper subvariety has relative congruence extension property if and only if it is a variety

Modal definability in enriched languages

Abstract The paper deals with polymodal languages combined with stan-dard semantics defined by means of some conditions on the frames. So a notion of "polymodal base " arises which provides various enrichments of the classical modal language. One of these enrichments, viz. the base £(R,-R), with modalities over a relation and over its complement, is the paper's main paradigm. The modal definability (in the spirit of van Benthem's correspon-dence theory) of arbitrary and ~-elementary classes of frames in this base and in some of its extensions, e.g., £(R,-R,R-1,_R-1), £(R,-R,=I=) etc., is described, and numerous examples of conditions definable there, as well as undefinable ones, are adduced. 81 Introduction Undoubtedly, first-order languages are reliable and universal tools for formalization. However, in some cases the cost of this universality is not fully acceptable: on the one hand we have the undecidability results, and on the other the fact that the expressive power of first-order languages does no

Complex algebras, varieties and games

Bibliography: leaves 123-126.Complex algebras have proven very useful in presenting the modern day logician with a tool to approach a wide variety of problems in the field of algebraic logic. This dissertation is intended as an exploration of various approaches to the study of complex algebras. In particular we will take a look at the logical and semantic views of complex algebras, as well as logical games involving these algebras

On chain domains, prime rings and torsion preradicals.

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Residually small varieties and commutator theory.

Thesis (M.Sc.)-University of Natal, Durban, 2000.Chapter 0 In this introductory chapter, certain notational and terminological conventions are established and a summary given of background results that are needed in subsequent chapters. Chapter 1 In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is introduced and used to characterize both subdirectly irreducible algebras and essential extensions. Special attention is paid to the role they play in varieties with definable principal congruences. The chapter focuses on residually small varieties; several of its results take their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture". One of the main results presented gives nine equivalent characterizations of a residually small variety; it is largely due to W. Taylor. It is followed by several illustrative examples of residually small varieties. The connections between residual smallness and several other (mostly categorical) properties are also considered, e.g., absolute retracts, injectivity, congruence extensibility, transferability of injections and the existence of injective hulls. A result of Taylor that establishes a bound on the size of an injective hull is included. Chapter 2 Beginning with a proof of A. Day's Mal'cev-style characterization of congruence modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"), this chapter is a self-contained development of commutator theory in such varieties. We adopt the purely algebraic approach of R. Freese and R. McKenzie [FM87] but show that, in modular varieties, their notion of the commutator [α,β] of two congruences α and β of an algebra coincides with that introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with the geometric approach proposed by Gumm [Gum80a],[Gum83]. Basic properties of the commutator are established, such as that it behaves very well with respect to homomorphisms and sufficiently well in products and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]” are proved. These results will be applied in the following chapters. We show how the theory manifests itself in groups (where it gives the familiar group theoretic commutator), rings, modules and congruence distributive varieties. Chapter 3 We define Abelian congruences, and Abelian and affine algebras. Abelian algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the greatest and least congruences of A). We show that an affine algebra is polynomially equivalent to a module over a ring (and is Abelian). We give a proof that an Abelian algebra in a modular variety is affine; this is Herrmann's Funda- mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78], [Gum80a] established that any modular variety has a so-called ternary "difference term" (a key ingredient of the Fundamental Theorem's proof). We derive some properties of such a term, the most significant being that its existence characterizes modular varieties. Chapter 4 An important result in this chapter (which is due to several authors) is the description of subdirectly irreducible algebras in a congruence modular variety. In the case of congruence distributive varieties, this theorem specializes to Jόnsson's Theorem. We consider some properties of a commutator identity (Cl) which is a necessary condition for a modular variety to be residually small. In the main result of the chapter we see that for a finite algebra A in a modular variety, the variety V(A) is residually small if and only if the subalgebras of A satisfy (Cl). This theorem of Freese and McKenzie also proves that a finitely generated congruence modular residually small variety has a finite residual bound, and it describes such a bound. Thus, within modular varieties, it proves the RS Conjecture. Conclusion The conclusion is a brief survey of further important results about residually small varieties, and includes mention of the recently disproved (general) RS Conjecture