$f$-Racks, $f$-Quandles, their Extensions and Cohomology

TL;DR

This paper introduces $f$-racks and $f$-quandles, generalizing classical algebraic structures by twisting identities, and develops their modules, extensions, and cohomology theories to expand their mathematical framework.

Contribution

It presents the first systematic study of $f$-racks and $f$-quandles, including constructions, classifications, and cohomology, extending the theory of racks and quandles.

Findings

Defined $f$-racks and $f$-quandles with examples

Classified low order $f$-quandles

Developed cohomology theory for $f$-quandles

Abstract

The purpose of this paper is to introduce and study the notions of $f$-rack and $f$-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order $f$-quandles. Moreover, we define modules over $f$-racks, discuss extensions and define a cohomology complex for $f$-quandles.