Re: [boost] thinking algebraically (was Re: proto::or_ arity

From: Marco (mrcekets_at_[hidden])
Date: 2007-04-22 12:28:54

On Sun, 22 Apr 2007 12:31:44 +0200, Larry Evans  
<cppljevans_at_[hidden]> wrote:

> Actually, I think and_ and or_ are the meet and join of
> a lattice:
>
> http://en.wikipedia.org/wiki/Lattice_%28order%29
>
> where:
>
>    proto::_
>      is the top    element
>    proto::not_<proto::_>
>      is the bottom element
>
> However, I'm not sure about this since I don't know
> if the and_ and or_ operations are associative
> and commutative, which is one of the requirements
> for a lattic (I think).
>

Well, the last word is up to the author's library, however I'd be  
surprised if the following relations were not valid:

associativity:
or_< T1, or_<T2, T3> > == or_< T1, T2, T3 > == or_< or_<T1, T2,>, T3 >
and_< T1, and_<T2, T3> > == and_< T1, T2, T3 > == and_< and_<T1, T2,>, T3 >

commutativity:
or_< T, U > == or_< U, T >
and_< T, U > == and_< U, T >

absorption:
or_< T, and_<T, U> > == T
and_< T, or_<T, U> > == T

obviously where "==" means logical equivalence not C++ data type equality.

Distibutivity shuold hold, too.

Marco

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