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)

Comments

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].

References

[a1]	E.L. Post, "Two-valued iterative systems of mathematical logic" , Princeton Univ. Press (1941)