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Omega-group

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Ω\Omega-groups

Idea

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

Definition

We take an algebraic structure in a traditional sense as a set Ω\Omega with a (not necessarily finite) number of operations α\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 Ω\Omega-group is an algebraic structure which amounts to a group (usually written additively but not necessarily commutative) together with a set Ω\Omega of operations of any arity, such that for each nn-ary operation αΩ\alpha \in \Omega, distributivity holds in each variable over the group operations:

α(x 1,,x j+y j,,x n)=α(x 1,,x j,,x n)+α(x 1,,y j,,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) .

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

Examples

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

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

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

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

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

Last revised on November 26, 2018 at 01:45:04. See the history of this page for a list of all contributions to it.