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\begin{document}

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\section*{Omega-group}

\hypertarget{groups}{}\section*{{$\Omega$-groups}}\label{groups}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{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 category|equationally presentable]] or [[algebraic category]].

An \textbf{$\Omega$-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:

\begin{displaymath}
\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) .
\end{displaymath}
(This states only distributivity over addition; however, distributivity over all other group operations follows.)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The classical examples are of course groups (where $\Omega$ is [[empty set|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 \textbf{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.\newline $\Omega$-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 \emph{category of interest}. (In the n-Lab we are trying out a perhaps more informative term\_ [[category of group-based universal algebras]]\_

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

The general theory of $\Omega$-groups is similar to the basics of group and ring theory, including [[normal subgroups]] / [[ideals]] / [[submodules]], [[quotient algebra|quotient]] $\Omega$-groups, Noether's [[isomorphism theorem]]s, etc. For example, the [[Jordan-Holder theorem|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; $\omega$-[[infinity-group|group]] is rather a synonym for (for some people strict) $\infty$-[[infinity-groupoid|groupoid]] with a single object, hence nothing to do with $\Omega$-groups.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \href{http://en.wikipedia.org/wiki/Group_with_operators}{Group with operators}.


\item N. Bourbaki, \emph{Algebra I}, ch. 1-3.


\item [[P. J. Higgins]], ``Groups with multiple operators'', Proceedings of the London Mathematical Society, 1956


\item E. I. Khukhro, \emph{Local nilpotency in varieties of groups with operators}, Russ. Acad. Sci. Sbornik Mat. 78 379, 1994. (\href{http://dx.doi.org/10.1070/SM1994v078n02ABEH003475}{doi})


\item Grace Orzech, \emph{Obstruction theory in algebraic categories I, II}, J. Pure Appl. Algebra \textbf{2} (1972) 287-340, 315--340.



\end{itemize}
[[!redirects Omega-group]] [[!redirects Omega-groups]] [[!redirects $\Omega$-group]] [[!redirects $\Omega$-groups]] [[!redirects ∞-group]] [[!redirects ∞-groups]]

[[!redirects group with operators]] [[!redirects groups with operators]]



\end{document}