Residually small varieties and commutator theory.

Abstract

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.