Representation of Algebraic Structures by Boolean Functions and Its Applications

Abstract

Boolean functions are mappings $\{0,1\}^n\rightarrow \{0,1\}$, where n is a nonnegative integer. It is well known that each Boolean function $f(x_1,\dots ,x_n)$ with n variables can be presented by its Algebraic Normal Form (ANF). If (G, F) is an algebra of order $|G|, \ 2^{n-1}\le |G|< 2^n$, where F is a set of finite operations on G, then any operation $f\in F$ of arity k can be interpreted as a partial vector valued Boolean function $f_{v.v.}:\{0,1\}^{kn}\rightarrow \{0,1\}^n$. By using the function $f_{v.v.}$ and ANF of Boolean functions, we can characterize different properties of the finite algebras, and here we mention several applications. We consider especially the case of groupoids, i.e., the case when $F=\{f\}$ consists of one binary operation and we classify groupoids of order 3 according to the degrees of their Boolean functions. Further on, we give another classification of linear groupoids of order 3 using graphical representation. At the end, we consider an application of Boolean representation for solving a system of equations in an algebra.