# Multi-operator group

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group with multiple operators, $\Omega$- group

A universal algebra which is a group relative to the addition operation $+$( which need not be commutative) and in which there is given a system of operations $\Omega$ of arity $\geq 1$. It is assumed that the zero element $0$ of the additive group $A$ is a subalgebra, that is, $0 \dots 0 \omega = 0$ for all $\omega \in \Omega$. Thus, a multi-operator group combines the concepts of a group, a linear algebra and a ring. An ideal of an $\Omega$- group is a normal subgroup $N$ of $A$ such that

$$- ( x _ {1} \dots x _ {n} \omega ) + ( x _ {1} \dots x _ {i-} 1 ( a + x _ {i} ) x _ {i+} 1 \dots x _ {n} \omega ) \in N$$

for all $a \in N$, $x _ {i} \in A$, $\omega \in \Omega$, $1 \leq i \leq n$. Congruences on a multi-operator group are described by coset classes relative to ideals.

Let $A$, $B$ and $C$ be $\Omega$- subgroups in an $\Omega$- group $G$( that is, subalgebras of the universal algebra $G$), where $C$ is generated by $A$ and $B$. The mutual commutator $[ A , B ]$ of the subgroups $A$ and $B$ is the ideal in $C$ generated by all elements of the form

$$- a - b + a + b ,$$

$$- ( a _ {1} \dots a _ {n} \omega ) - ( b _ {1} \dots b _ {n} \omega ) + ( a _ {1} + b _ {1} ) \dots ( a _ {n} + b _ {n} ) \omega ,$$

where $a , a _ {i} \in A$, $b , b _ {i} \in B$, $\omega \in \Omega$. Let $G ^ { \prime } = [ G , G ]$. A multi-operator group $G$ is called Abelian if $G ^ { \prime } = 0$. Inductively one defines ideals $G _ {i+} 1 = [ G _ {i} , G ]$, where $G _ {1} = G ^ { \prime }$, and $G ^ {(} i+ 1) = [ G ^ {(} i) , G ^ {(} i) ]$, where $G ^ {(} 1) = G ^ { \prime }$. A multi-operator group $G$ is called nilpotent if $G _ {i} = 0$, and solvable if $G ^ {(} i) = 0$ for some $i$. Many of the properties of the corresponding classes of groups and rings can be transferred to these classes of multi-operator groups. A multi-operator group $A$ is called a multi-operator (linear) $\Omega$- algebra over a commutative associative ring $k$ with an identity if the addition in $A$ is commutative, if $\Omega _ {1} = k$, where $\Omega _ {1}$ is the set of unary operations from $\Omega$, and if all operations from $\Omega$ are semi-linear over $k$( see [2][6], and Semi-linear mapping).

#### References

 [1] P.J. Higgins, "Groups with multiple operators" Proc. London Math. Soc. , 6 (1956) pp. 366–416 [2] A.G. Kurosh, "Free sums of multi-operator algebras" Sibirsk. Mat. Zh. , 1 : 1 (1960) pp. 62–70 (In Russian) [3] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) [4] A.G. Kurosh, "General algebra. Lectures for the academic year 1969/70" , Moscow (1974) (In Russian) [5] A.G. Kurosh, "Multioperator rings and algebras" Russian Math. Surveys , 24 : 1 (1969) pp. 1–13 Uspekhi Mat. Nauk. , 24 : 1 (1969) pp. 3–15 [6] T.M. Baranovich, M.S. Burgin, "Linear -algebras" Russian Math. Surveys , 30 : 4 (1975) pp. 65–113 Uspekhi Mat. Nauk. , 30 : 4 (1975) pp. 61–106 [7] V.A. Artamonov, "Universal algebras" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 191–248 (In Russian) [8] , Rings , 1 , Novosibirsk (1973) pp. 41–45
How to Cite This Entry:
Multi-operator group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-operator_group&oldid=47920
This article was adapted from an original article by V.A. Artamonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article