Nonstandard varieties of topological algebraic automata
by
T.M. Otryvankina
Saratov State University, Russia

In this note the Birkhoff theorem about structural characterzation of varieties of algebraic automata [1, theorem 12.1] is generalized to topological algebraic automata with the help of methods of the nonstandard analysis [2].

By a topological algebraic automaton (TAA for short) we mean a two-sorted algebra A=(G, S, o ), where G is a topological semigroup, S is a topological algebra of a type T and a binary operation o : S×G --> S coordinated with algebraic operations and topologies.

By analogy with [3] to describe properties of TAA we use a nonstandard two-sorted formal language L over sets X and Z. Nonstandard terms of the language L are elements of nonstandard extensions ^*W(X) and ^*W₁(Z), where W(X) is the free semigroup over X and W₁(Z) is the T₁-algebra of T₁-words over Z for the type T₁=T U  W(X). Formulas of L are defined on induction by usual way. An identity t₁=t₂ is said to be a topological if it hasn't symbols of the type T and semistandard if one of terms t₁, t₂ is standard.

A class of TAA K is said to be a nonstandard (accordingly semistandard or topological) variety if it is axiomatizable by a class of nonstandard (accordingly semistandard or topological) identities.

Theorem. Let K be a class of TAA closed under the formation of direct products. Then the following statements hold:

(i) K is a nonstandard variety if and only if it is closed under the formation of closed subsistems and continuous homomorphic images;

(ii) K is a standard variety if and only if it is closed under the formation of subsistems and homomorphic images;

(iii) K is a semistandard variety if and only if it is closed under the formation of closed subsistems and continuous homomorphic images;

(iv) K is a topological variety if and only if it is closed under the formation of closed subsistems and continuous images.

REFERENCE

[1]  Plotkin B.I., Gringlaz L.Y., Gvaramiya A.A., Elements of algebraic automata theory. - M. Vysshaya shkola, 1994.

[2]  Devis M.H.A., Applied Nonstandard Analysis , Wiley & Sons. New York, 1977.

[3]  Molchanov V.A., Nonstandard varieties of pseudotopological algebraic systems // Sibirskii matematicheskii zhurnal, V. 32 (1991), No. 3, P. 104-112.

Date received: March 16, 2000