# Regularity of bounded tri-linear and the fourth adjiont of tri-derivation

## TL;DR

This paper establishes a criterion for the regularity of bounded tri-linear mappings and explores conditions under which the fourth adjoint of a tri-derivation remains a tri-derivation.

## Contribution

It provides a simple criterion for regularity of tri-linear mappings and characterizes when the fourth adjoint of a tri-derivation is also a tri-derivation.

## Key findings

- A necessary and sufficient condition for the regularity of a tri-linear mapping.
- Characterization of when the fourth adjoint of a tri-derivation is a tri-derivation.
- A criterion linking the regularity of a tri-linear map to the regularity of its composition with a bounded linear map.

## Abstract

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such the fourth adjoint $D^{***}$ of a tri-derivation $D$ is again tri-derivation.