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Boolean transformations with unique fixed points. (English) Zbl 1199.06055
Let \(B\) be a Boolean algebra, \(n\) an integer \(>1\) and \(f\:B^n\to B^n\). \(f\) is called a Boolean respectively simple Boolean transformation, if its component functions are polynomial respectively term functions. Boolean transformations having a unique fixed point are characterized. Moreover, it is shown that in case \(n=2\) there exist exactly \(36\) simple Boolean transformations having exactly one fixed point and that \(12\) of them are isotone.

MSC:
06E30 Boolean functions
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[1] BROWN F. M.-RUDEANU S.: Consequences, consistency and independence in Boolean algebras. Notre Dame J. Formal Logic 22 (1981), 45-62. · Zbl 0423.03057
[2] CHILEZAN C.: Some fixed point theorems in Boolean algebra. Publ. Inst. Math. (Beograd) 28(42) (1980), 77-82.
[3] LÖWENHEIM L.: Gebietdeterminanten. Math. Ann. 79 (1919), 222-236.
[4] PARKER W. L.-BERNSTEIN B. A.: On uniquely solvable Boolean equations. Univ. Calif. Publ. Math. NS 3(1) (1955), 1-29.
[5] ROUCHE N.: Some properties of Boolean equations. IRE Trans. Electronic Comput. EC-7 (1958), 291-298.
[6] RUDEANU S.: Boolean Functions and Equations. North-Holland, Amsterdam, 1974. · Zbl 0321.06013
[7] RUDEANU S.: On the range of a Boolean transformation. Publ. Inst. Math. (Beograd) 19 (33) (1975), 139-144. · Zbl 0326.02043
[8] RUDEANU S.: Injectivity domains of Boolean transformations. Rev. Roumaine Math. Pures Appl. 23 (1978), 113-119. · Zbl 0377.02035
[9] RUDEANU S.: Fixpoints of lattice and Boolean transformations. An. String Univ. Al. I. Cuza Iasi. Mat. 26 (1980), 147-153.
[10] RUDEANU S.: Lattice Functions and Equations. Springer-Verlag, London, 2001. · Zbl 0984.06001
[11] WHITEHEAD A. N: Memoir on the algebra of symbolic logic. Amer. J. Math. 23 (1901), 297-316. · JFM 32.0388.03
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