12
$\begingroup$

I am studying Hungerford's book "Algebra". In the page 27 he defines the meaningful product as follows.

Given any sequence of elements of a semigroup $G, > \{a_{1},a_{2},\dots\}$ define inductively a meaningful product (in this order) as follows. If $n=1$, then the only meaningful product is $a_{1}$. If $n>1$, then a meaningful product is defined to be any product of the form $(a_{1}\cdots a_{m})(a_{m+1}\cdots a_{n})$ where $m< n$ and $(a_{1}\cdots a_{m})$ and $(a_{m+1}\cdots a_{n})$ are meaningful products of $m$ and $n-m$ elements respectively.

He notes next the following:

To show that this definition is in the fact well defined requires a stronger version of Recursion Theorem 6.2 of the Introduction; see C.W. Burril: Foundations of Real Numbers.

I don't have access to this book, so I would like to know this version and see how to use it, or a reference if possible.

I've never seen this definition before. Is it really necessary to define a meaningful product in order to prove that Generalized Associative law holds on a semigroup?

Thanks for your help.

$\endgroup$
  • $\begingroup$ The recursion theorem he is alluding to is almost surely the standard wellfounded recursion theorem. At any rate, a meaningful product of length $n$ is easily seen to be the same thing as a binary tree with $n$ leaves... $\endgroup$ – Zhen Lin Jan 27 '12 at 0:35
  • $\begingroup$ He does it for a monoid, and I'm lazy enough not to check if it works for a semigroup, but Jacobson (Basic Algebra, Vol. 1, § 1.4, pp. 39-40) seems to prove generalized associativity. $\endgroup$ – Bruno Stonek Jan 29 '12 at 0:34
2
$\begingroup$

What is often done for semigroups is defining positive integer powers of elements, and defining products using associativity of multiplication in the semigroup.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.