From pre-trusses to skew braces

TL;DR

This paper introduces the concept of pre-trusses, explores their properties, and establishes connections to skew braces and near-rings, including the development of quotient structures and fraction constructions.

Contribution

It defines pre-trusses and near-trusses, studies their congruences and structures, and links these to skew braces and near-rings, advancing algebraic theory.

Findings

Pre-trusses are sets with heap and semigroup structures.

Near-trusses and skew trusses are characterized by distributive laws.

Skew braces correspond to near-trusses of fractions without an absorber.

Abstract

The notion of a pre-truss, that is, a set that is both a heap and a semigroup is introduced. Pre-trusses themselves as well as pre-trusses in which one-sided or two-sided distributive laws hold are studied. These are termed near-trusses and skew trusses respectively. Congruences in pre-trusses are shown to correspond to paragons defined here as sub-heaps satisfying particular closure property. Near-trusses corresponding to skew braces and near-rings are identified through their paragon and ideal structures. Regular elements in a pre-truss are defined leading to the notion of a (pre-truss) domain. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are defined. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.