# Zhegalkin algebra

The special algebra , where

and is the multiplication operation. The clone of the action of on is of interest. Every operation in is a polynomial , 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 is contained in . Thus, the study of properties of includes, in particular, the study of all algebras for arbitrary .

#### 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 or the free Boolean algebra on 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 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) |

**How to Cite This Entry:**

Zhegalkin algebra. V.B. Kudryavtsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Zhegalkin_algebra&oldid=11979