nLab Omega-group

Idea

An $\Omega$-group is a group equipped with additional algebraic operations (of signature $\Omega$) that distribute over the group operations.

Definition

We take an algebraic structure in a traditional sense as a set $\Omega$ 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 $\alpha \in \Omega$, distributivity holds in each variable over the group operations:

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

(This states only distributivity over addition; however, distributivity over all other group operations follows.)

Examples

The classical examples are of course groups (where $\Omega$ is empty), rngs (where $\Omega$ consists of only multiplication), and rings (where $\Omega$ consists of multiplication and the nullary operator that gives the multiplicative identity). Given a fixed ground ring $k$, the modules over $k$ form another example: each element of $k$ gives a unary multiplication operation.

The older term group with operators is traditionally used for $\Omega$-groups when only unary operations are considered (as in the case of modules).

Philip Higgins discussed a more general notion of $\Omega$-group: a group $G$ endowed with a set $\Omega$ of finitary operations satisfying the condition that the neutral element of the group should form a one-element subalgebra.

$\Omega$-groups in the sense of Higgins form a protomodular category, which is not in general strongly protomodular. A counterexample is provided by the category of digroups, i.e. sets with two group group structures that share the same neutral element.

Grace Orzech introduced a notion of extension in a special related setting known, somewhat opaguely, as a category of interest. (In the n-Lab we are trying out a perhaps more informative term_ category of group-based universal algebras_

Remarks

The general theory of $\Omega$-groups is similar to the basics of group and ring theory, including normal subgroups / ideals / submodules, quotient $\Omega$-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 $\Omega$-groups is also interesting.

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

References

• Wikipedia, Group with operators.

• N. Bourbaki, Algebra I, ch. 1-3.

• P. J. Higgins, “Groups with multiple operators”, Proceedings of the London Mathematical Society, 1956

• 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.