About a certain class of non-associative algebra. Aroev Dilshod Davronovich

About a certain class of non-associative algebra.

Aroev Dilshod Davronovich

Kokand State Pedagogical Institute

PhD, senior lecturer

Tel .: [phone], [e-mail]

Sobirov Avazbek Abdurashid oglu

Kokand State Pedagogical Institute

2nd year student majoring in mathematics teaching methods

Annotation

In this paper, a class of non-associative algebras is constructed in a certain way and their properties are studied.

Keywords: reflection, isomorphic reflection, displacement, cycle, cycle length.

let the collection be given. If the reflection is mutually exclusive, two are optional in the set

identified between the elements

The set with respect to the + and * operations forms a non-associative loop. Here * is the product of the product and the corresponding coordinate product of s. Let’s mark it with a ring. following in the ring

the practice of multiplying a number by equality can also be determined, i.e., the set forms an associative algebra over the operations mentioned.

If there is a reciprocal one-valued reflection that satisfies the equality for and two mutually equivalent reflections, then and the reflections are said to be similar.

of each element

in appearance.

apparently the sum of the elements is determined.

multiplication in the set

can be expressed in terms of equality.

In relation to these actions, the set forms an associative ring.

Andy

the product of two elements using reflection

if we define it equally, this action does not obey the law of associativity, but fulfills the condition of distributiveness, i.e. the actions given in the set satisfy the following properties.

here

For any element, there is an opposite element that satisfies the equation:

.

Let us denote by the ring defined by the above conditions.

Theorem: and for algebras to be isomorphic and substitutions to be similar, it is necessary and sufficient.

Proof: Necessity. and whether it is possible to construct an isomorphic reflection between algebras. and show that the substitutions are similar. of space using isomorphism

it is obvious that the base passes into the base of space. In addition, the unit is converted to another vector using a vector isomorphism. Let the equations be reasonable. In that case put in place

in appearance the equation holds for the belly and the substitutions, i.e., and the substitutions are similar.

Sufficiency. and that the substitutions are similar, i.e., that there is a substitution that satisfies the equality.

let us define the reflection by equality. In that case the reflection will have a reciprocal value because it is a substitution.

from the last equation it follows that the reflection + practice is preserved. We now show that the reflection * saves the action.

There are three types of "multiplication" and the operation is defined in algebra (the operation is multiplication by coordinates. So it is an isomorphic reflection.

References.

1. A.G.Kurosh. Kurs vysshey algebry // Nauka, M.1975.-S. 572.

2. S.N.Nasirov. Description of idealov odnogo klassa neassotsiativnyx algebr // Nauchnye trudy TashGU, vyp 418, 1972. –S. 232-236

3. S.N.Nasirov. About the ideals of a single class of neo-social algebra // Nauchnye trudy, TashGU, vyp 461, 1974. –S. 86-89.