Multi-operator group

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