By A.G. Pinus

During the previous few many years the information, tools, and result of the idea of Boolean algebras have performed an expanding function in a number of branches of arithmetic and cybernetics.

This monograph is dedicated to the basics of the speculation of Boolean buildings in common algebra. additionally thought of are the issues of offering assorted forms of common algebra with those structures, and functions for investigating the spectra and skeletons of sorts of common algebras.

For researchers whose paintings contains common algebra and good judgment.

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**Example text**

21. Any countable commutative semigroup is isomorphically embeddable into < JBtko; x>, which is a countable Cartesian skeleton of a variety of Boolean algebras. For an algebraic description of a countable Cartesian skeleton of a variety of Boolean algebras a number of definitions will be required. 39. (a) A commutative monoid < M; ',1> is called canonical if the equality xy = 1 yields the equalities x = Y = 1. (b) A general refinement of the sequences < xi,l i ~ n >< Yjlj ~ m> of the elements of a commutative monoid with the property Xi = Yj is a sequence n n isn < r;}i ~ n,j ~ m> such that for any jsm i ~ n,j ~ m (c) The monoid < M;;l > is called a refinement monoid if it is canonical and any sequences of the elements of M < Xiii ~ n >, < Yjlj :s m> with the property Xi = Yj have a general refinement.

E. Dobbertin [53]. B q are non-isomorphic when the ultrafilters p,q are incomparable relative to the Rudin-Keisler order. Negrepontis [41]. Pinus [178] (their proofs are presented in section 14 in Applications). 2. Basic Notions of Universal Algebra The purpose of this section is to recall the basic facts of the theory of universal algebras to be used later in this monograph. Sankappanavar [28] and others, will be made use of. Let us recall some of them. For any class of algebras R, I(R) will denote the class of all algebras isomorphic to the algebras of the class R .

Pn variables such that for i = O, ... 'Ul Po(X,y,z) = x, Pn(x,y,z) = Z, Pi(X,y,x) = x; Pi(X,X,Z) = of three : Pi+l(x,x,Z) for even i; Pi(x,Z,Z) = Pi+l(x,Z,Z) for odd i. 10. 'Ul equations: p(X,y,X) satisfies the following = p(x,y,y) = p(y,y,x) = x. It should be noted, for example, that any variety of groups, rings and quasi- groups is congruence-commutable. 8, one can also observe that any congruence-commutable variety is congruence-modular, and any variety of lattices is congruence-distributive.