## 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.