[FOM] three variables are needed for defining BA

[Istvan Nemeti]

This is Re: Re: 629: Boolean algebra/Simplicity (Harvey Friedman), where he
writes: 

 

"Incidentally, all of the axiomatizations of BA that I have seen use three
variables. Must be a classic result that you can't get away with 2
variables?"

 

Indeed, this is the case. It is proved in [Diamond, A. H. and McKinsey, J.
C. C., Algebras and their subalgebras, Bull. Amer. Math. Soc. 53 (1947),
147-154] that the class of all Boolean algebras cannot be defined by two
variables. The proof goes by showing that there is an algebra  which is not
a Boolean algebra but each of its two-generated subalgebras  is a Boolean
algebra.  In [Quackenbush, R. W., Near-Boolean algebras I: Combinatorial
aspects, Discrete Mathematics 10 (1974), 301-308], a finite algebra with
these properties is given.

 

Best regards,  Hajnal Andreka and Istvan Nemeti

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