## Abstract

An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side we call it a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [8]. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [14]. Congruences in pre- and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragons under which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology, pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.

Original language | English |
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Pages (from-to) | 683-714 |

Number of pages | 32 |

Journal | Publicacions Matematiques |

Volume | 66 |

DOIs | |

Publication status | Published - 1 Jul 2022 |

## Keywords

- heap
- near-ring
- near-truss
- pre-truss
- skew brace

## ASJC Scopus subject areas

- General Mathematics