# Zhegalkin algebra

The special algebra $\mathfrak A = \langle A , \Omega \rangle$, where

$$A = \{ 0 , 1 \} ,\ \ \Omega = \{ {x \cdot y } : {x + y \ ( \mathop{\rm mod} 2 ) , 0 , 1 } \} ,$$

and $x \cdot y$ is the multiplication operation. The clone $F$ of the action of $\Omega$ on $A$ is of interest. Every operation in $F$ is a polynomial $\mathop{\rm mod} 2$, a so-called Zhegalkin polynomial, named after I.I. Zhegalkin, who initiated the investigation of this clone [1]. He proved that every finitary operation on $A$ is contained in $F$. Thus, the study of properties of $F$ includes, in particular, the study of all algebras $\mathfrak A = \langle A , \Omega ^ \prime \rangle$ for arbitrary $\Omega ^ \prime$.

#### References

 [1] I.I. Zhegalkin, Mat. Sb. , 34 : 1 (1927) pp. 9–28 [2] P.M. Cohn, "Universal algebra" , Reidel (1986) [3] S.V. Yablonskii, G.P. Gavrilov, V.B. Kudryavtsev, "Functions of the algebra of logic and Post classes" , Moscow (1966) (In Russian)

In other words, the Zhegalkin algebra is the two-element Boolean ring, the field $\mathbf Z /( 2)$ or the free Boolean algebra on $0$ generators. As such, it is generally not given a distinctive name in the Western literature. Cf. e.g. Boolean algebra; Boolean equation. The study of all algebras $\mathfrak A = \langle A, \Omega ^ \prime \rangle$ is the subject of E.L. Post's dissertation [a1].