By A.G. Pinus

During the previous couple of a long time the information, tools, and result of the idea of Boolean algebras have performed an expanding function in a variety of branches of arithmetic and cybernetics.

This monograph is dedicated to the basics of the idea of Boolean buildings in common algebra. additionally thought of are the issues of providing diversified different types of common algebra with those buildings, and purposes 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**

A) Boolean algebras B(Fa) are Bonnet-rigid; (b) for a < bER, B(Fa)«B(Ib). B , has the order type of a set of real numbers. Let us give an example of constructing rigid Boolean algebras. Shelah [207], who constructed large families of mutually unembeddable models of not supers table theories, which are based on constructing corresponding families of trees.

II y) V (x II z), A lattice L is called modular if it satisfies the following equality: (x II y) V (y II z) = Y II «x II y) v z), or a quasi-equality equivalent to it: X II Y= X- X V (y II z) = y II (x v z). Any distributive lattice is modular. 6. A lattice L is non-modular iff a lattice Ns is isomorphically imbeddable in L (Fig. 3). 7. A lattice L is non-distributive iff a lattice M3 or isomorphically imbeddable into L (Fig. 3). Ns is <) Fig. 3 A variety of algebras such that Con21 is distributive (modular) for all its algebras 21 is called a congruence-distributive (congruence-modular) variety.

Blb. B I a into any ).. Bj(i < 2 ) of the power A such that for any i". B,l a into any homomorphic image of the algebraB}lb. 17 yield, in particular, the following corollary. 7. Bi(i E2A) of Boolean algebras of the power A which are mutually unembeddable and are not homomorphic images of each other. 7 for singular A such that 2~o < A, and A~o = A. 17 is Bonnet-rigid and, in particular, has no injective endomorphism onto itself. B1 such that 11 ". fz. If It is not injective, then there is an a EB such that a".