This exposition is the third volume of a three-volume series of books, and consists of Chapters 9, 10 and 11. Chapters 1–5 are included in Volume I [R. N. McKenzie et al., Algebras, lattices, varieties. Volume I. With an additional bibliography. Reprint of the 1987 original published by Wadsworth & Brooks/Cole Advanced Books & Software. Providence, RI: AMS Chelsea Publishing (2018; Zbl 1392.08001)], and Chapters 6–8 in Volume II [R. S. Freese et al., Algebras, lattices, varieties. Volume II. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1509.08001)].
Chapter 9 (“Finite algebras and their clones”) begins with the definitions of a clone of operations on a set and related concepts (essentially \(k\)-ary element, essential spectrum of a clone, etc.). Next, the Galois connection associated with the relation of preservation between operations and relations over finite sets is examined. The third section discusses narrow spectra of some classes of algebras (a set \(S\) of positive integers is called narrow if the set of primes \(p\) with the property (\(\exists n\in S\))(\(p\) divides \(n\)) is finite), including the variety generated by a paraprimal algebra (an algebra P is called paraprimal provided it is finite, there is a Mal’tsev term \(m(x,y,z)\) so that P \(\models m(x, x, y) \approx y\approx m(y,x, x)\), and every nontrivial subalgebra of P is simple). Separate sections are devoted to: the lattice of clones on a 2-element set; primal, quasiprimal, functionally complete, hereditarily simple algebras; Rosenberg’s theorem on the description of proper clones and maximal clones; theorems of V. L. Murskiǐ and Roy O. Davies on calculating the probabilities for a finite algebra to have a given algebraic property; clones where all operations satisfy the term condition (an \(n\)-ary operation \(F\) satisfies the term condition if \(F(a_0, \dots, a_{i-1}, x, a_{i+1},\dots, a_{n-1})=F(b_0, \dots, b_{i-1}, x, b_{i+1},\dots, b_{n-1})\) implies \(F(a_0, \dots, a_{i-1}, y, a_{i+1},\dots, a_{n-1})=F(b_0, \dots, b_{i-1}, y, b_{i+1},\dots, b_{n-1})\) for all \(i, x, y\) and \(a_i\)). The last section of the chapter discusses the problems associated with finding the free spectrum of a variety.
Chapter 10 (“Abstract clone theory”) begins with a definition of abstract clones and homomorphisms, followed by a number of examples of clones (of operations on sets, of an algebra, of a variety, of a monoid, of a ring with unit etc). Separate sections are devoted to representations of clones by varieties and by categories; presentation of clones by graded sets and clone-theoretic equations over these sets (establishing a connection between presentations of clones and axiomatizations of varieties); interpretations and functors between varieties; products, coproducts and ultraproducts of clones; power varieties and algebras \(A^{[k]}\) (=\(A\times\dots\times A\) with an additional unary operation \(c(\langle a_0,\dots,a_{k-1}\rangle)=\langle a_{k-1}, a_0,\ldots, a_{k-2}\rangle\) and a \(k\)-ary operation \(d(\langle a_0^0,\dots,a_{k-1}^0\rangle,\dots, \langle a_0^{k-1},\dots,a_{k-1}^{k-1}\rangle) =\langle a_0^0,\dots,a_{k-1}^{k-1}\rangle\)); category equivalence of varieties; lattice of classes of varieties based on the interpretability relation (called the interpretability lattice); ideals and filters of interpretability lattice and Mal’tsev-definable classes of varieties. The last section introduces hypervarieties and brings several examples of them.
Chapter 11 (“Commutator theory”) begins with definitions of weak centrality, centrality, strong centrality and strong rectangularity binary relations and corresponding algebras (weakly abelian algebra, abelian algebra, etc.). Next, commutators and annihilators are introduced, the main properties are derived, and examples are provided. Let \(S, T\) be tolerance relations on an algebra A. The commutator \([S,T]\) of \(S\) and \(T\) is the least congruence \(\delta\) such that \(f(\mathbf{a,u})\delta f(\mathbf{a,v})\Longleftrightarrow f(\mathbf{b,u})\delta f(\mathbf{b,v})\), where \(f(\mathbf{x,y})\) is an \((m+n)\)-ary polynomial of A, \(\mathbf{a}S\mathbf{b}\) and \(\mathbf{u}T\mathbf{v}\) (componentwise). Centrality relations and commutators are described in congruence modular varieties. The relationships between sixteen commutator properties of varieties are discussed in detail. These properties include: \([\alpha,\beta]=[\beta,\alpha]\); \([\bigvee\alpha_i,\beta]=\bigvee[\alpha_i,\beta]\); congruence modular; \(k\)-permutability for some \(k\); congruence meet semidistributive; congruence join semidistributive; congruence neutral (\([\alpha,\beta]=\alpha\wedge\beta\) for all congruences); all abelian algebras are affine; having a difference term (a ternary term \(d(x,y,z)\) in the signature of a variety \(\mathcal{V}\) is called a difference term for \(\mathcal{V}\) if it is idempotent and for each \(\mathbf{A}\in \mathcal{V}\), for every \(a,b\in A\) and each congruence \(\theta\) on \(\mathbf{A}\) such that \(a\theta b\), \(d^{\mathbf{A}}(a,a,b)=b\) and \(d^{\mathbf{A}}(a,b,b)[\theta,\theta]a\)); etc. One section is devoted to nilpotent and solvable algebras, abelian varieties and to the ring of a variety. In separate sections, residual smallness and finite axiomatizability in congruence modular varieties are discussed. In the last section, locally finite varieties that have some additional properties (for example, abelian congruence modular or varieties that have, up to isomorphism, only finitely many finite directly indecomposable algebras) is considered.
The text is provided with additional explanations, historical comments and open problems. Most sections are provided with several exercises.
There are numerous references in the text to what is discussed in Volumes I and II. At the same time, the detailed proofs of the results are presented, making Volume III self-contained.