nLab
Omega-group

An algebraic structure is here taken in a traditional sense as a set SS 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, the distributivity in each variable holds:

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)

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 Ω\Omega-groups when only unary operations/operators are considered. The general theory of Ω\Omega-groups is similar to the basics of group and ring theory, including ideals, quotient Ω\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 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.

  • 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.

Revised on June 26, 2010 15:50:09 by John Baez (99.11.157.15)