From: magidin@math.berkeley.edu (Arturo Magidin)
Subject: Re: universal algebra
Date: Tue, 4 Sep 2001 14:14:31 +0000 (UTC)
Newsgroups: sci.math
Summary: Extending universal algebra to ordered categories

In article <20010904004746.B21669-100000@agora.rdrop.com>,
William Elliot  <marsh@agora.rdrop.com> wrote:
>The notion of an algebra can include
>	sets, lattices, groups, rings, lattice groups, vector spaces, etc.
>Has this notion been extended to
>	(partially) ordered sets, groups, etc.

I assume you mean partially-ordered groups, partially ordered rings, etc.

The order is not an operation, it's a relation. But you can extend the
notion of algebras to include partial operations and relations, as
well as operations. This has been done.

However, many of the "nice" results of algebra fail when you have
relations and partial operations. For example, under this definition,
an "Algebra" would be a set, together with a family of total
operations, a family of partial operations, and a family of
relations. A subalgebra would be one in which the induced relation
holds, the operations are closed, and any tuple in the subalgebra in
the domain of the partial operation has a result in the subalgebra.

But now the image of a morphism need not be a subalgebra of the image.

>Can the notion of an algebra be extended to include a topological space?

There are (rather complicated) ways of trying to do this. They tend to
obscure, rather than help, though.

>It seems to me that any mathematical structure that can be reduced with an
>equivalence relation will have a corresponding form of the Fundamental
>Homomorphism Theorem.

Unfortunately, you run into problems once you get partial operations
and relations. The Homomorphism Theorem has to be rephrased to make it
still hold.

>So my question is basically how universal algebra for functions or
>operations over domains can be extended to set structures and
>relationships?  Can this be done with characteristic functions or is there
>a more natural way of doing this?  Of particular interest are binary order
>relations, positive cones of groups, etc and topologies or collections of
>special (open) sets.

A "type" is defined as an ordered triple, (F, P, R), where each of F,
P, R are ordered tuples; F contains the "function"/operator symbols, together
with their arity; P contains the "partial operator" symbols, together
with their arity; R contains relations on F, toegher with their arity.

An algebra of type T is a set X, together with: for every f in F of
arity a, a map f:F^a->F.

For every p in P of arity a, a map p:X->F, where X is a subset of F^a.

For every r in R of arity a, a subset r of F^a.

(Note that you can reduce all types to relations if you want, but it
is often easier to think of the types as having operations, partial
operations, and relations).

An "identity" would be defined similarly as usual, except that when it
involves partial operations or relations then it only holds "when
defined."

A morphism respects operations (usual meaning); respects partial
operations: whenever (a_1,...,a_n) lies in the domain of p,
f(a_1),....,f(a_n) lies in the domain of the corresponding partial
operator p', and the image of p(a_1,...,a_n) is p' applied to
(f(a_1),...,f(a_n)). And respects relations: whenever (a_1,...,a_n)
lies in r, (f(a_1),...,f(a_n)) lies in the corresponding relation r'
of the image.

    [.snip.]

======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu