An introduction to loopoids

TL;DR

This paper introduces loopoids, a non-associative generalization of groupoids, and explores their properties and applications, particularly in the context of Lagrangian discrete mechanics.

Contribution

It presents the concept of loopoids and semiloopoids, expanding the algebraic framework for discrete mechanics beyond traditional groupoids.

Findings

Defined and studied properties of loopoids and semiloopoids

Proposed a differential version of loopoids for mechanics applications

Established foundational concepts for non-associative algebraic structures

Abstract

We discuss a concept of loopoid as a non-associative generalization of (Brandt) groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.