Talk:Boolean algebra (structure)
Latest comment: 1 month ago by Jochen Burghardt in topic $1$ and $0$ are supposed to be distinct in the definition
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About isomorphisms
editIn § 4, Homomorphisms and isomorphisms, why's isomorphisms being mentioned in the title, but not in the section itself? Are all homomorphisms also isomorphisms? That should be written then.
--Unknowledgeable (talk) 13:50, 18 July 2016 (UTC)
- I've added the definition of isomorphism; not every homomorphism is an isomorphism. Andrewbt (talk) 05:45, 31 December 2017 (UTC)
Anti-absorption
editAnother useful identity (theorem, not axiom) is:
- a ∨ (¬a ∧ b) = a ∨ b
- a ∧ (¬a ∨ b) = a ∧ b
Does this have a standard name? It follows immediately from distributivity, of course. Should it be mentioned in the article? --Macrakis (talk) 20:09, 26 March 2021 (UTC)
$1$ and $0$ are supposed to be distinct in the definition
editI think it is typical (See for instance page 10 of "Introduction to Boolean algebras" by Paul Halmos) that in the definition of the algebra, $0$ and $1$ are distinct elements. Should I make the change? PierreQuinton (talk) 13:59, 20 September 2024 (UTC)
- In English, "two elements 0 and 1" means "two" not "one or two". D.Lazard (talk) 14:21, 20 September 2024 (UTC)
- I'd say it's a little ambiguous. I would say "two distinct elements" if that's definitely what we mean. (I think it's a slightly controversial point whether there's a one-element Boolean algebra; in my usage there is not, but I'm not sure there's a complete consensus on it in the literature.) --Trovatore (talk) 19:16, 20 September 2024 (UTC)
- We have a remark about that in section Definition: "
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.)
" - Moreover, as far as I remember, it makes a big difference if the trivial one-element boolean algebra is to be excluded: an extra axiom is needed, since it is not an equation, the class is no longer a variety, hence Birkhoff's variety theorem no longer applies to it. That might be the reason why the distinctness requirement is missing in newer works. - Jochen Burghardt (talk) 05:20, 23 September 2024 (UTC)
- I see that that is marked "citation needed". The "older works" thing is potentially especially problematic; are we sure that there aren't newer works that still require 0 and 1 to be distinct? --Trovatore (talk) 15:19, 23 September 2024 (UTC)
- I'm not sure about newer works. What about just omitting "
In older works
", for now? - Jochen Burghardt (talk) 15:45, 29 September 2024 (UTC)
- I'm not sure about newer works. What about just omitting "
- I see that that is marked "citation needed". The "older works" thing is potentially especially problematic; are we sure that there aren't newer works that still require 0 and 1 to be distinct? --Trovatore (talk) 15:19, 23 September 2024 (UTC)
