nLab
Omega-group

An algebraic structure is here taken in a traditional sense as a set $S$ with a (not necessarily finite) number of operations $\alpha$ 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 $\Omega$-group is an algebraic structure which amounts to a group (usually written additively but not necessarily commutative) together with a set $\Omega$ of operations of any arity, such that for each $n$-ary operation $\alpha \in \Omega$, the distributivity in each variable holds:

$$\alpha (x_1,\dots ,x_j+y_j,\dots ,x_n)=\alpha (x_1,\dots ,x_j,\dots ,x_n)+\alpha (x_1,\dots ,y_j,\dots ,x_n)$$\alpha(x_1,\ldots,x_j+y_j,\ldots,x_n)= \alpha(x_1,\ldots,x_j,\ldots,x_n)+\alpha(x_1,\ldots,y_j,\ldots,x_n)

The classical examples are of course groups and rings, but also modules over a fixed ring: each element of the ground ring is a unary operation. An older term group with operators is traditionally used for $\Omega$-groups when only unary operations/operators are considered. The general theory of $\Omega$-groups is similar to the basics of group and ring theory, including ideals, quotient $\Omega$-groups, isomorphism theorems, 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 $\Omega$-groups is also interesting.

Note that $\Omega$ is a capital Greek letter; $\omega$-group is rather a synonym for (for some people strict) $\mathrm{\infty }$-groupoid with a single object, hence nothing to do with $\Omega$-groups.

• 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](http://dx.doi.org/10.1070/SM1994v078n02ABEH003475)]

• Grace Orzech, Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2 (1972) 287-340, 315–340.