Re: Semigroup problem

On Mon, 12 May 2008, William Elliot wrote:

On Sun, 11 May 2008 marcjhg@xxxxxxxxx wrote:

F and G are functions of arity 1. * is function of arity 2. I will use
infix notation for *. (% will be a symbol for comment)

The axioms are:

all x all y exists z (x * y = z). % Closure
all x all y all z (x * (y * z) = (x * y) * z). % Associativity
all z exists x exists y (x * y = z). % Surjectivity
all x all y (F(x) * G(y) = y * x).

Assume semigroup with identity e and left and right cancellation.

f(e) g(y) = y
f(x) g(e) = x

f(e) g(e) = e = ee = f(e) g(e) f(e) g(e)
e = g(e) f(e)

For the same result, instead of assuming
left and right cancellation, just assume
e = g(e) f(e)

In particular, if f = g and the semigroup is a monoid
(ie, has an identity), then symmetry follows.

xy = f(x) g(e) f(e) g(y) = f(x) g(y) = yx

On the ground of these axioms only try to prove:
all x all y (x * y = y * x). % Commutativity

Or try to find a model (finite if itRs possible) in which holds:
exists x exists y (x * y != y * x). % Non-commutativity


.

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