Nonassociative right hoops

TL;DR

This paper introduces nonassociative right hoops (narhoops), a new algebraic structure that generalizes right quasigroups and right hoops, with detailed characterization of their properties and subvarieties.

Contribution

It defines narhoops as a new variety of nonassociative right-residuated magmas and explores their algebraic properties, subvarieties, and congruence relations.

Findings

Narhoops form a variety of algebras.

Subvarieties with associative and/or commutative operations are characterized.

Congruences are determined by the class of the left unit.

Abstract

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation $x\sqcap y=(x / y)y$ is associative and/or commutative. Narhoops with a left unit are proved to have a top element if and only if $\sqcap$ is commutative, and their congruences are determined by the equivalence class of the left unit. We also show that the four identities defining narhoops are independent.