Re: Transitive Closure

From: x <x-false_at_yahoo.com>
Date: Tue, 18 May 2004 10:32:59 +0300
Message-ID: <40a9bb5a$1@post.usenet.com>

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"Mikito Harakiri" <mikharakiri_nospaum_at_yahoo.com> wrote in message news:8a529bb.0405171058.36a5a26_at_posting.google.com...
> Paul <paul_at_test.com> wrote in message

news:<Hu2qc.4347$wI4.496108_at_wards.force9.net>...
> > x wrote:
> > >>The commutative property does not make sense with unary operators like
> > >>TClose. Unary operators are always commutative because there is only
> > >>one operand's order.
> > >
> > >
> > > Commutativity does make sense with unary operators.
> > > M--f-->M--g-->M
> > > fg = gf or fg != gf
> >
> > I think the confusion here is that in something like group theory we say
> > a binary operator "*" is commutative if a*b = b*a. Or writing it as a
> > function we say:
> >
> > *(a,b) = *(b,a)
> >
> > What we're talking about here which I think is what Alfredo is
> > misunderstanding is commutativity of the "composition of operators"
> > operator. So say we have unary operators f and g we can define the
> > operator "f*g" to be:
> >
> > (f*g)(x) = f(g(x)) for all x.
> >
> > Commutativity of this "*" operation just means that:
> >
> > f*g = g*f
> >
> > ie. f(g(x)) = g(f(x)) for all x.
> >
> > For this to make sense f and g must take arguments of the same type, and
> > return values of that same type as well.

> I'm confused myself.

> So we are talking about different algebra, that has a universe with
> elements being unary relational algebra operators only -- selection,
> projection, Cartesian Power (as product is, unfortunately, binary
> operator), transitive closure, and, perhaps, negation. In this tiny 4
> element universe we define one binary operation: composition of unary
> relational operators. So we can speak of this operation commutativity,
> although, it's kind of wierd to say that operation commutes on these 2
> particular elements of the universe, and doesn't commute on those. In
> traditional algebra commutativity is all or nothing proposition: it's
> either commutative on the whole universe, or isn't (if there is at
> least a pair of elements such that ab!=ba).

> Can people familiar with Category Theory comment? (Diagram by "x"
> provoked this thought).

I'm not familiar with Category Theory but...

com-mu-ta-tive (kuh myue'tuh tiv, kom'yuh tay tiv) adj.

of or pertaining to commutation, exchange, substitution, or interchange.
a. (of a binary operation) having the property that one term operating on a second is equal to the second operating on the first, as a x b = b x a.
having reference to this property: the commutative law for multiplication.

A binary operation such multiplication: x can be viewed as a family of unary operators fa(u)=a x u
Then a x b = fa(fb(1)) = fb(fa(1))
Commutativity (of symbols) means interchangeability (of symbols).

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Received on Tue May 18 2004 - 02:32:59 CDT