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In Wikipedia, the Boolean algebra is defined as a 6-tuple $(A,\wedge,\vee,\neg,0,1)$. In Kuratowski1976, on the other side in the definition on page 34, there is no $1$. Halmos1963 has the $1$.

Does the definition in Kuratowski1976 leads to something more general or somehow different theory? On page 37 he introduces the concept of unit $i$ that I assume is the $1$ of the other authors (also judging from the definition $a\wedge i=a$).

Definition from Kuratowski1976:

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Mohamed Ali
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2 Answers2

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All Boolean algebras have a top element.

I don't have access to that Kuratowski definition, but tipically, what happens is these alternative definitions have axioms enough to define a partial order relation relative to which the complement operation is order-reversinng.

Given that $0$ is the bottom element, it will follow that $0'$ is the top one.

amrsa
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  • I've added the definition from the source. – Mohamed Ali Jun 25 '19 at 04:51
  • Is it really possible from the given axioms to get the element $1$? The two operations are not the complement, am I wrong? Then the complement is not part of the axioms. – Mohamed Ali Jun 25 '19 at 04:54
  • @AaronLenz If your goal is try to derive other properties from that set of axioms, then I think you should make another question asking for that explicitly. I can't think immediately how to define $\vee$ and $'$ from those axioms, but it must be possible, since the concept of Boolean algebra is not ambiguous. – amrsa Jun 25 '19 at 08:55
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I have no access to Kuratowski's definition currently, but I assume that he might want to define something less restrictive (depending on whether all the other axioms are the usual ones or equivalent) that we would usually not call boolean algebra and afterwards introduce the notion of $1$ and what we call boolean algebras. That is basically the same as the different notions for rings. There are associative rings, associative unitary rings, commutative rings and what I personally call a ring (commutative unitary). Depending on the context some notion might be better, appears more often or is of main interest such that it makes sense to choose a fitting definition.

Edit: As amrsa pointed out: In the case that really only the $1$ element is missing, then you should just be able to get it as the complement of $0$ with the other axioms.

Con
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