On the diagrams of classes of conditionally termal functions
by
Alexander Pinus
Novosibirsk State Technical University

Works [1]-[5] define classes of conditionally termal (CT(A)), elementary conditionally termal (ECT(A)), positive conditionally termal (PCT(A)) and existential positive conditionally termal ( exists⁺CT(A)) functions on any universal algebra A. They also give some descriptions of these functions with the help of some Galois connections. For any algebra A we have the following diagram:

T(A) subset or equal PCT(A) ^{subset or equal exists+CT(A) subset or equal} _{subset or equal CT(A) subset or equal} ECT(A). (*)

Here T(A) is the class of all termal functions on the algebra A.

Let Aut(A), End(A), Iso(A), Ihm(A) are group of all automorphisms, the semigroup of all endomorphisms, the semigroup of all innere isomorphisms, the semigroup of all innere homomorphisms of algebra A correspondently, and End₁(A)={g in End(A)| |rang g|=1}, Ihm₁(A)={g in Ihm(A)| |rang g|=1}. We study the conditions for which we have the equations in the diagram (*).

1. For any finite algebra A

exists⁺CT(A)=ECT(A) <===> End(A)=Aut(A) \cup End₁(A),

PCT(A)=CT(A) <===> Ihm(A)=Iso(A) \cup Ihm₁(A).

If any innere homomorphism of algebra A can be extended to some endomorphism of algebra A, then we denote this fact as Exihm(A).

2. For any finite algebra A:

Exihm(A) ===> exists⁺CT(A)=PCT(A).

There exists some finite algebra A such that exists⁺CT(A)=PCT(A)& not Exihm(A).

3. For any finite endoprimal algebra A we have equation exists⁺CT(A)=T(A). There exists some finite non-endoprimal algebra A such that exists⁺CT(A)=T(A).

Theorem. 1).The equation T(A)=PCT(A) is independent from any other equations in the diagram (*). 2).For any finite algebra A : exists⁺CT(A)=CT(A) ===> CT(A)=ECT(A), PCT(A)=CT(A) ===> exists⁺CT(A)=ECT(A).

The other equations on the right side of the diagram (*) are pairwise independent.

References.

1. A.G.Pinus. On the conditional terms and identities on universal algebras. //Siberian Adv. Math., v.8, No.2, 1998, p.96-109.

2. A.G.Pinus. The calculus of conditional identities and conditional rational equivalence. //Algebra and logic, v.37, No.4, 1998, p.432-459.

3. A.G.Pinus. N-elementary embeddings and n-conditional terms. //Mathematica. Izvestiya VUZov, No.1, 1999, p.36-40.

4. A.G.Pinus. On the functions which commutes with the semigroups of transformations of algebras. //Siberian Institute of Mathematics, v.41, No.6, 2000, p.1409-1418.

5. A.G.Pinus. Innere homomorphisms and positive conditional terms. //Algebra and logic, v.40, No.1, 2001.

Date received: January 18, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagi-05.