# Semi-Associative $3$-Algebras

## TL;DR

This paper introduces semi-associative 3-algebras, explores their structures, modules, and extensions, and establishes their connection to 3-Lie algebras, expanding the understanding of non-associative algebraic systems.

## Contribution

It defines semi-associative 3-algebras, constructs their modules and extensions, and links them to 3-Lie algebras, providing new frameworks and tools for non-associative algebra research.

## Key findings

- Defined semi-associative 3-algebras and their properties.
- Constructed double modules and extensions using cocycles.
- Established the relationship between semi-associative 3-algebras and 3-Lie algebras.

## Abstract

A new 3-ary non-associative algebra, which is called a semi-associative $3$-algebra, is introduced, and the double modules and double extensions by cocycles are provided. Every semi-associative $3$-algebra $(A, \{ , , \})$ has an adjacent 3-Lie algebra $(A, [ , , ]_c)$. From a semi-associative $3$-algebra $(A, \{, , \})$, a double module $(\phi, \psi, M)$ and a cocycle $\theta$, a semi-direct product semi-associative $3$-algebra $A\ltimes_{\phi\psi} M $ and a double extension $(A\dot+A^*, \{ , , \}_{\theta})$ are constructed, and structures are studied.