Quandles are distributive algebraic structures that were introduced by David Joyce [24] in his Ph.D. dissertation in 1979 and at the same time in separate work by Matveev [34]. Quandles can be used to construct invariants of the knots in the 3-dimensional space and knotted surfaces in 4-dimensional space. Quandles can also be studied on their own right as any non-associative algebraic structures. In this dissertation, we introduce f-quandles which are a generalization of usual quandles. In the first part of this dissertation, we present the definitions of f-quandles together with examples, and properties. Also, we provide a method of producing a new f-quandle from a given f-quandle together with a given homomorphism. Extensions of f-quandles with both dynamical and constant cocycles theory are discussed. In Chapter 4, we provide cohomology theory of f-quandles in Theorem 4.1.1 and briefly discuss the relationship between Knot Theory and f-quandles. In the second part of this dissertation, we provide generalized 2,3, and 4- cocycles for Alexander f-quandles with a few examples. Considering “Hom-algebraic Structures” as our nutrient enriched soil, we planted “quandle” seeds to get f-quandles. Over the last couple of years, this f- quandle plant grew into a tree. We believe this tree will continue to grow into a larger tree that will provide future fruit and contributions.

Quandles are distributive algebraic structures originally introduced independently by David Joyce and Sergei Matveev in 1979, motivated by the study of knots. In this dissertation, we discuss a number of generalizations of the notion of quandles. In the first part of this dissertation we discuss biquandles, in the context of augmented biquandles, a representation of biquandles in terms of actions of a set by an augmentation group. Using this representation we are able to develop a homology and cohomology theory for these structures. We then introduce an n-ary generalization of the notion of quandles. We discuss a number of properties of these structures and provide a number of examples. Also discussed are methods of obtaining n-ary quandles through iteration of binary quandles, and obtaining binary quandles from n-ary quandles, along with a classification of low order ternary quandles. We build upon this generalization, introducing n-ary f-quandles, and similarly discuss examples, properties, and relations between the n-ary structures and their binary counter parts, as well as low order classification of ternary f-quandles. Finally we present cohomology theory for general n-ary f-quandles.