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Zhegalkin algebra

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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
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098