Pat, (01)
> >Not following, the definitions in the math links say all elements
> >in the range have mappings from some element in the domain, while
> >the co-domain can have elements that aren't mapping to by any
> >element in the domain.
>
> Yes. But at least the issue is about sets and
> relations. The definitions for 'range' in the
> wikipedia CS entry (pointer above) are to do with
> boundary values in arrays and limits on values of
> variables, which are a COMPLETELY different
> topic. I thought this was your point. (02)
I take variables as properties (in the OWL sense) of processes as they
are executing (like the processes you see in the task list of operating
systems). Their types are the ranges (in the OWL sense) of those
properties. (03)
> >In OWL, if the range of a property pet is Animal, it isn't
> required that all animals are pets. (04)
> Right, so the owl:range is a subset of the co-domain, as I said. (05)
I think it;s the (Wiki) co-domain: (06)
http://en.wikipedia.org/wiki/Codomain
(Wolfram doesn't give a term for this concept, as far as I can tell). (07)
The co-domain is the second element in the cross product that defines
the tuples of the relation. The domain is the first. The
(Wiki/Wolfram) range is are all and only those elements in the co-domain
that are mapped to from some element in the domain. (08)
In the above example, the pet property has co-domain Animal, and its
range is the Pet class (qua-type in KL-one). (09)
> In fact, if S is any set which is a subset of the codomain and which
> has the math-range as a subset, then it would be correct to assert
> that S is an owl-range for the property. (010)
Sure, you can always limit the co-domain to the range without ill
effect, but the typically the co-domain is used as a way of specifying
what individuals *can* be values of the property, which the range
doesn't tell you. I've heard of approaches that always use the range as
the co-domain, and assume "reclassification" of an individual as a
member of the range class is restricted by some other means. (011)
> >Not the way I've heard idempotence used by software people.
> >They're referring to calling the same function twice on the same
> >arguments (the wikipedia ambiguously calls this "used multiple
> >times"), where the second call has no effect. (012)
> Ah, I see your point, sorry. Hmm. I think I see where this
> happened. If you assume that every application involves a 'state',
> then multiple applications become nested function applications to a
> starting state, which maps the computer science sense into the
> mathematical sense. (013)
And even then you need to assume the CS functions return the system
state, which of course, they don't. (014)
> >I thought it equivalent to a set of disjoint classes that are also
> a union for another. What's the mathematical definition? (015)
> It unions distinct 'copies' of the sets, so even if they are
> identical or share elements, one gets distinct copies in the
> disjoint union (aka 'separated union'). This can be described
> formally in various ways, but the key property is that the
> cardinality of the disjoint union is always the sum of the
> cardinalities of the unioned sets. So the n-fold disjoint union of A
> with itself is meaningful and isomorphic to A x n. If the sets are
> disjoint, then the disjoint union is the ordinary union. But it is
> defined even if they aren't disjoint; and this is important for the
> category-theory treatment of set theory. (016)
I see, the category people are assuming bags. But union can apply to
sets, and the individuals conforming to an OWL class are a set, aren't
they? (017)
BTW, I remembered a case of being overly careful in borrowing
terminology. The UMLers decided to use the term Multiplcity instead of
Cardinality for the restrictions on the number of values of a property,
on the grounds that "cardinality" means the number of values in set,
whereas what is needed is a constraint on those. This causes
unfortunate disconnects when UMLers talk with database modelers and
OWLers. (018)
Conrad (019)
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