On Sat, 29 Oct 2005, Christopher D. Malon wrote:
For the non-mathematically inclined: A field is a set with two binary operations, + and *. Under either operation (+ or *), the set is an abelian (= commutative) group, and a field has a distributive property: a * (b + c) = a*b + a*c. An easy example is the set of real numbers, with + being addition and * being multiplication.
Not exactly. With the exception of _the_ trivial case (up to a natural isomorphism there's only one!) a field is _not_ a multiplicative group wrt its multiplication, given that the identity for the addition can't have a multiplicative inverse.
This is of some relevance of so called Universal Algebra in which some commonly used structures (e.g. a group, a ring) can be described in terms of sets of n-ary operations obeying a set of algebraic identities, whereas certain other just as common structures (e.g. a field) can't - and a more complex approach is needed.
This may be of some interest here too, since there's been some talking about the algebra of methods of a role that any class which C<does> that role guarantees to obey by doing so. So this should take into account the possibility of algebrai relations "not defined everywhere", which may bring a new level of complexity in realistic situations...
Michele -- I'm finding now in my 40s that the less makeup I wear, the better. I think softer is better as you get older. With everything. Except men. - Kim Cattrall
