[FOM] three variables are needed for defining BA
Istvan Nemeti
inemeti at gmail.com
Sun Oct 4 09:48:48 EDT 2015
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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