Multi-operator group
group with multiple operators, -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
of arity
. It is assumed that the zero element
of the additive group
is a subalgebra, that is,
for all
. Thus, a multi-operator group combines the concepts of a group, a linear algebra and a ring. An ideal of an
-group is a normal subgroup
of
such that
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for all ,
,
,
. Congruences on a multi-operator group are described by coset classes relative to ideals.
Let ,
and
be
-subgroups in an
-group
(that is, subalgebras of the universal algebra
), where
is generated by
and
. The mutual commutator
of the subgroups
and
is the ideal in
generated by all elements of the form
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where ,
,
. Let
. A multi-operator group
is called Abelian if
. Inductively one defines ideals
, where
, and
, where
. A multi-operator group
is called nilpotent if
, and solvable if
for some
. 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
is called a multi-operator (linear)
-algebra over a commutative associative ring
with an identity if the addition in
is commutative, if
, where
is the set of unary operations from
, and if all operations from
are semi-linear over
(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 ![]() |
[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 |
Multi-operator group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multi-operator_group&oldid=47920