# What is the “Boolean algebra fragment of RA”?

The Wikipedia article on Relation Algebra notes that this is a formal system which has essentially the same expressive power as the three-variable fragment of first-order logic. Peano Arithmetic can be encoded in the three-variable fragment, so this means that the satisfiability of an arbitrary RA formula is undecidable. There is an unsourced but intriguing remark:

(N.B. The Boolean algebra fragment of RA is complete and decidable.)

What is meant by "the Boolean algebra fragment of RA"?

I have tried to track down references, but I don't have access to the Tarski and Givant book which is one of the main texts (and which seems relevant judging by Monk's 1989 review). Any hints, references, or pointers (especially to journal articles or lecture notes that are online) would be welcome.

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I think this might be more suitable for MO. – Kaveh Apr 30 '11 at 20:54
@Kaveh: perhaps so, though I suspect the answer is quite simple for someone who know it. – András Salamon Apr 30 '11 at 21:20
My guess would be what the first sentence of the WP article says, but I guess you have seen that already: "a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.", i.e. formulas restricted to the Boolean part. – Kaveh Apr 30 '11 at 21:23
@Kaveh: yes, that would be my guess too. I'm hoping for a reference or a definite statement, since I would like to build on this result -- a guess is not my favourite kind of foundation. – András Salamon Apr 30 '11 at 22:24
I think you may post it as a reference request question, if your guess is wrong they will tell you. :) – Kaveh Apr 30 '11 at 22:32