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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.

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  • $\begingroup$ 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
    Commented Jan 27, 2012 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. $\endgroup$ Commented Jan 29, 2012 at 0:34

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

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