# Definition of meaningful product

1. Feb 15, 2012

### gotjrgkr

1. The problem statement, all variables and given/known data
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,>{a1,a2,…} define inductively a meaningful product (in this order) as follows. If n=1, then the only meaningful product is a1. If n>1, then a meaningful product is defined to be any product of the form (a1⋯am)(am+1⋯an) where m<n and (a1⋯am) and (am+1⋯an) 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?