By A. I. Mal’cev (auth.)

As some distance again because the 1920's, algebra have been accredited because the technology learning the houses of units on which there's outlined a selected method of operations. notwithstanding up till the 40s the overpowering majority of algebraists have been investigating in simple terms a couple of forms of algebraic buildings. those have been basically teams, earrings and lattices. the 1st basic theoretical paintings facing arbitrary units with arbitrary operations is because of G. Birkhoff (1935). in the course of those related years, A. Tarski released an enormous paper within which he formulated the elemental prin ciples of a conception of units built with a method of kinfolk. Such units are actually referred to as types. unlike algebra, version conception made abun dant use of the gear of mathematical good judgment. the opportunity of making fruitful use of common sense not just to check common algebras but in addition the extra classical components of algebra reminiscent of crew thought used to be dis lined through the writer in 1936. through the subsequent twenty-five years, it steadily grew to become transparent that the speculation of common algebras and version concept are very in detail comparable regardless of a definite distinction within the nature in their difficulties. And it really is for that reason significant to talk of a unmarried concept of algebraic platforms facing units on which there's outlined a sequence of operations and kin (algebraic systems). The formal equipment of the speculation is the language of the so-called utilized predicate calculus. hence the idea should be thought of to frame on good judgment and algebra.

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Models and Algebras 43 where not only all systems 2il> 2i 2, ••• coincide but also all isomorphisms q;j. Consider for example the algebra 2iI = ({1, 2, 3, ... J; +). The mapping q;: a -+ 2a is an isomorphic mapping of 2iI onto its sub algebra (2, 4, 6, ... J and so the chain of isomorphisms (3) forms a direct spectrum. The question is: What is the limit algebra of the spectrum? In order to distinguish the elements of the 2iI located in the n-th position of the chain (3) from those of the algebra located in other positions, we shall denote the elements of the n-th algebra 2iI by the pairs (n, 1), (n, 2), ...

A partial ordering of a set A is said to be a linear (total, simple) ordering if every pair of elements a and b in A are comparable under ;2;. Let ;2; be a partial ordering of a set A and let B ~ A, An element a E A is an upper bound for B in A if b ;2; a for all b E B. An element a E A is said to be greatest in A if a is an upper bound for A itself. An element mEA is said to be maximal in A if each element x in A is not comparable with m or x ;2; m. If a set A has a greatest element a, then a will obviously be the only maximal element in A.

All others are called transfinite cardinals. Let a = IA I and b = IBI. We define a ~ b if there exists a one-to-one mapping of A into B. It is easy to show that the definition is independent of the choice of A and B and so expresses a relation between cardinal numbers. Theorem 1. The relation ~ is a linear ordering in any set of cardinal numbers. From the definition of ~, it follows that this relation is reflexive and transitive. Its antisymmetry is a consequence of the following CantorBernstein theorem.