On nonstandard methods in the theory of pseudovarieties
by
Vladimir Molchanov
Saratov State University, Saratov, Russia

Using the nonstandard approach to the model theory we give in [4] a nonstandard characterization of some well-known classes of algebraic systems which can not be axiomatized by formulas of the lower predicate language and investigate a nonstandard construction of free algebraic systems over SP_fin-closed class of finite algebraic systems. These results make it possible to investigate the Eilenberg's correspondence between pseudovarieties of finite \Omega-algebras and varieties of recognizable languages with the help of nonstandard congruencies on the nonstandard extension ^*W of the \Omega-algebra W = W_\Omega(X) over a countable set X.

Based on the nonstandard analysis methods we obtain also the following compactness theorem.

Theorem. Let B be an \Omega-algebra, F a family of congruencies on B directed by set inclusion and F_i   (i=1, ..., k) families of internal subsets of ^*B directed by set inclusion. Then for any positive formula j(x₁, ..., x_n, y₁, ..., y_k) of the lower predicate language of the algebraic type \Omega the following conditions are equivalent:

1) for every \epsilon in F, F_i in F_i  (i=1, ..., k) the factor-algebra B/\epsilon satisfies the property


( for allx1, ..., xn)( existsy1 in F1, ..., yk in Fk) j(x1, ..., xn, y1, ..., yk);


2) there exist b₁, ..., b_k in ^*B such that b_i in \cap { ^*F_i : F_i in F_i }  (i=1, ..., k) and for the congruence \mu = \cap { ^*\epsilon: \epsilon in F} the factor-algebra ^*B/\mu satisfies the property


( for allx1, ..., xn) j(x1, ..., xn, \mu(b1), ..., \mu(bk)).

Applications of this result to the theory of pseudovarieties are considered.

References

[1]  J. Almeida, On pseudovarieties, varieties of languages, filters of congruences, pseudoidentities and related topics , Algebra Universalis 27 (1990) 333-350.

[2]  V.A.Molchanov, Nonstandard characterization of pseudovarieties , Algebra Universalis, 1995, V. 33. P. 533-547.

[3]  S. Albeverio, J. E. Fenstad, R. Höegh-Krohn and T. L. Lindstrøm Nonstandard methods in stochastic analysis and mathematical physics , Academic Press, New York, 1986.

[4]  V.A.Molchanov, On nonstandard axiomatization of elementary non-axiomatizable classes of algebraic systems , Sibirskii matematicheskii zhurnal, 1999, V.40, No.2. P. 421-433 (in Russian).

Date received: May 11, 2004