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

Communicated by S. López-Permouth

Keywords:

AMSC: 17A30, 20G10

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