In this paper we present a brief study of the $σ$-set-$σ$-antiset duality that occurs in $σ$-set theory and we also present the development of the integer space $3^{A}=\left\langle 2^{A}, 2^{A^{-}} \right\rangle$ for the cardinals $|A|=2,3$ together with its algebraic properties. In this article, we also develop a presentation of some of the properties of fusion of $σ$-sets and finally we present the development and definition of a type of equations of one $σ$-set variable.