On the proper automorphisms of universal algebras
by
Alexander Pinus
Novosibirsk State Technical University

We consider automorphisms of universal algebras which are defined using the universal algerba itself. We say that an automorphism j of algerba A is termal (conditionally termal, elementary conditionally termal) if there exists a term (conditional term, elementary conditional term) t(x) of algebra A such that j(a)=t(a) for any element a from A. We say that an automorphism j of algebra A is polynomial (conditionally polynomial, elementary conditionally polynomial) if there exist a term (conditional term, elementary conditional term) t(x, y₁, ... , y_n) and elements b₁, ... , b_n from A so that j(a)=t(a, b₁, ... , b_n) for any element a from A. The automorphism j of algebra A is purely proper (purely conditionally proper, purely elementary conditionally proper, proper, conditionally proper, elementary conditionally proper) if j is represented as product of a purely termal (purely conditionally termal, purely elementary conditionally termal, termal, conditionally termal, elementary conditionally termal) automorphisms of algebra A. We denote the group of all purely proper (purely conditionally proper, purely elementary conditionally proper, proper, conditionally proper, elementary conditionally proper) automorphisms of algebra A as SP Aut(A) (CSP Aut(A), ECSP Aut(A), P Aut(A), CP Aut(A), ECP Aut(A)). Let Z(G) be the center of a group G. Let also Iso(A) be the semigroup of inner isomorphisms of algebra A.

Theorem 1: For any group G, for any normal subgroup G₁ of the group G and for any subgroup G₂ of the group G such that G₂ is subset of the intersection of G₁ and Z(G) there exists an universal algebra A and an isomophism h of group G on the group Aut(A) such that h(G₁)=P Aut(A) and h(G₂)=SP Aut(A).

Theorem 2: a). Let A be the uniformly locally finite algebra of finite signature (finite algebra of any signature). Then for any automorphism j of algebra A the following conditions are equivalent:

1). j is from CSP Aut(A).

2). All subalgebras of algebra A are j-closed and j commutes with every function from Iso(A).

b). For any finite algebra A and for any j from Aut(A) the following conditions are equivalent:

1). j is from ECSP Aut(A).

2). All subalgebras of algebra A are j-closed and j is from Z(Aut(A)).

Theorem 3: For any uniformly locally finite algebra A of finite signature and for any j from Aut(A) the following conditions are equivalent:

1). j is from CP Aut(A).

2). There exists a finite subalgebra A₁ of algebra A such that all subalgebras L of algebra A so that A₁ is a subset of L are j-closed and j commutes with any h from Iso(A) such that A₁ is a subset of Dom(h).

Date received: January 17, 2003