# Associative Protomodular Algebras

## TL;DR

This paper introduces a new notion of associativity for high-arity operations and proves that a variety of universal algebras contains a group operation if and only if it contains an associative semi-abelian operation.

## Contribution

It defines a novel concept of associativity for high-arity operations and establishes a characterization of when a variety includes a group operation.

## Key findings

- A new notion of associativity for high-arity operations is introduced.
- A characterization of varieties containing group operations is provided.
- The algebraic theory's structure is linked to semi-abelian operations with this new associativity.

## Abstract

The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the algebraic theory of a variety of universal algebras contains a group operation if and only if it contains a semi-abelian operation which is associative in the sense introduced.