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From: gwyn@brl-tgr.ARPA (Doug Gwyn )
Newsgroups: net.math
Subject: Re: Beyond Exponention
Message-ID: <8072@brl-tgr.ARPA>
Date: Wed, 6-Feb-85 02:06:53 EST
Article-I.D.: brl-tgr.8072
Posted: Wed Feb  6 02:06:53 1985
Date-Received: Sun, 10-Feb-85 06:40:27 EST
References: <460@decwrl.UUCP>
Organization: Ballistic Research Lab
Lines: 18

> but like the 2 preceding functions a+b and a*b, a$b is both communitive
> and associative, that is a$b==b$a and a$(b$c)==(a$b)$c.  It also obeys
> the distributive law, that is a$(b*c)==(a$b)*(a$c).

If you want to get algebraic, observe that a*b is not a group
(no inverse for 0) and in general one expects to lose an algebraic
characteristic for each operation in the progression.  This is a lot
like the progression real -> complex -> quaternion.

I don't think there is a single "right" answer to the question,
"What operation is the third in the sequence +, *, ...?" any more
than there is a single right answer to "What is the fourth number in
the sequence 3, 5, 7, ...?"

George Terell told me once that he had investigated the problem
of making a generalized series of +, *, ... operators when he was
in high school.  I never got any details, though.  George was last
seen by me at the Houston Medical Center and/or Rice campus.