# Trusses: Paragons, ideals and modules

## TL;DR

This paper explores the algebraic structure of trusses, introducing ideals, paragons, and modules, and provides a classification of truss structures on integers, bridging ring theory and module theory.

## Contribution

It introduces the concepts of ideals, paragons, and modules in trusses, and classifies truss structures on the integers, expanding algebraic theory.

## Key findings

- Classification of truss structures on integers
- Definition of modules over trusses
- Conditions for sub-heap to induce module structure

## Abstract

Trusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several constructions of trusses are presented. A full classification of truss structures on the Abelian group of integers is given. Modules over trusses are defined and their basic properties and examples are analysed. In particular, the sufficient and necessary condition for a sub-heap of a module to induce a module structure on the quotient heap is established.