Информационная система "Конференции"

Abstracts

Model theory and set theory

In [1,2] the additive and primitive connected theories were studied. The example of the same theories is the module theory. At that, additive theories are the special case of primitive connected theories and additive theories just as module theories are primitive normal [1]. Here we study S-polygons with primitive normal and additive theories.

Let S be monoid. A (left) S-polygon is a set on which S acts unitarity from the left in usual way. By S-Act we denote the class of all S-polygons. Class K of S-polygons is called primitive normal (additive, primitive connected) if the theory of each S--polygon from K is primitive normal (additive, primitive connected) accordingly. Monoid S is called linearly ordered if {Sa | a is from S} is linearly ordered set relatively inclusion.

Theorem 1. Class S-Act is primitive normal iff S is linearly ordered monoid.

In [3] it is note that class S-Act is primitive connected iff S is group. But there are nothing monoids with additive class S-Act, that is we obtain the follow

Theorem 2. Class S-Act is not additive for any monoid S.


References

[1] Palyutin E.A. Additive theories, in: Proceedings of Logic Colloquim'98 (Lectures Notes in Logic, 13), ASl, Massachusetts, 2000, 352--356.

[2] Palyutin E.A. Primitive connected theories // Algebra and Logic -- 2000.-- V. 39, № 2.

[3] Stepanova A.A. Primitive connected poligons classes // International conference on algebra, logic and cybernetics. -- Irkutsk, 2004 (in Russian).

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