Subject: categories: Re: Laws

From: [email-masked]
Date: Tue, 8 Aug 2006 02:30:20 -0400 (EDT)

To respond to Leinster's inquiry,

"Laws" (or "equations"), as the set-based universal
algebraists understand them, are ordered pairs of
members of free algebras (i.e., pairs e = (e_1, e_2)
in F x F, for F an algebra free on some set of "free
generators."

Actually, far more often than not, the variety of
algebras these F are free in is presented by means
of operations only, and the F are then called
"absolutely free."

A given equation e "holds" in an algebra A with the
given operations iff under each homomorphism from F
to A the elements e_1 and e_2 of F are shipped to
some same value in A.

>From this perspective the Abelianness equation xy=yx
is the pair (xy, yx) in F2 x F2 (F2 denoting the
absolutely free algebra on the two free generators
x & y based on, say, three operations, one binary
(multiplication), one unary (inversion), one nullary
(choice of base point).

The associativity equation x(yz) = (xy)z is another
equation in this sense.

One need not, of course, insist dogmatically on
taking as equations ONLY pairs in absolutely free
algebras: no harm in considering pairs in free
algebras of any variety. Thus, for example,
(xy, yx) is still a reasonable equation for groups.
But (x(yz), (xy)z) doesn't do what you think:
the RHS and LHS are ALREADY equal in every group,
and the pair is simply the diagonal entry (xyz, xyz)
(the INTENDED associativity is already a FACT for
groups, not, like commutativity, a condition that,
capable of failing, may meaningfully be imposed).

If these comments don't fully address the concerns raised,
please let me know. In any event, the laws most UAers
speak of refer to equations in absolutely free algebras
coming from the "lawless" variety whose algebras use the
same operations as another variety one is more interested
in, but are subject to the imposition of no equations
at all.

-- Fred

Tom Leinster had written:

> Dear category theorists,
>
> Here's something that I don't understand.  People sometimes talk about
> algebraic structures "satisfying laws".  E.g. let's take groups.  Being
> abelian is a law; it says that the equation xy = yx holds.  A group G
> "satisfies no laws" if
>
>     whenever X is a set and w, w' are distinct elements of the free
>     group F(X) on X, there exists a homomorphism f: F(X) ---> G
>     such that f(w) and f(w') are distinct.
>
> For example, an abelian group cannot satisfy no laws, since you could take
> X = {x, y}, w = xy, and w' = yx.  There are various interesting examples
> of groups that satisfy no laws.
>
> To be rather concrete about it, you could define a "law satisfied by G" to
> be a triple (X, w, w') consisting of a set X and elements w, w' of F(X),
> such that every homomorphism F(X) ---> G sends w and w' to the same thing.
>  A law is "trivial" if w = w'.  Then "satisfies no laws" means "satisfies
> only trivial laws".
>
> You could then say: given a group G, consider the groups that satisfy all
> the laws satisfied by G.  (E.g. if G is abelian then all such groups will
> be abelian.)  This is going to be a new algebraic theory.
>
> What bothers me is that I feel there must be some categorical story I'm
> missing here.  Everything above is very concrete; for instance, it's
> heavily set-based.  What's known about all this?  In particular, what's
> known about the process described in the previous paragraph, whereby any
> theory T and  T-algebra G give rise to a new theory?
>
> Thanks,
> Tom
>
>
>
>