We take an algebraic structure in a traditional sense as a set with a (not necessarily finite) number of operations of various arity and satisfying some axioms (not necessarily of first order). That is, we are discussing objects of an equationally presentable or algebraic category.
An -group is an algebraic structure which amounts to a group (usually written additively but not necessarily commutative) together with a set of operations of any arity, such that for each -ary operation , distributivity holds in each variable over the group operations:
(This states only distributivity over addition; however, distributivity over all other group operations follows.)
The classical examples are of course groups (where is empty), rngs (where consists of only multiplication), and rings (where consists of multiplication and the nullary operator that gives the multiplicative identity). Given a fixed ground ring , the modules over form another example: each element of gives a unary multiplication operation.
The older term group with operators is traditionally used for -groups when only unary operations are considered (as in the case of modules).
The general theory of -groups is similar to the basics of group and ring theory, including normal subgroups / ideals / submodules, quotient? -groups, Noether’s isomorphism theorem?s, etc. For example, the Jordan–Holder theorem holds: if there is a composition series, then every two composition series are equivalent up to permutation of factors. An obvious horizontal categorification of -groups is also interesting.
-groups form a protomodular category.
Wikipedia, Group with operators.
N. Bourbaki, Algebra I, ch. 1-3.
E. I. Khukhro, Local nilpotency in varieties of groups with operators, Russ. Acad. Sci. Sbornik Mat. 78 379, 1994. (doi)
Grace Orzech, Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2 (1972) 287-340, 315–340.