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*To*: categories@mta.ca*Subject*: categories: Re: Real interval halving*From*: Vaughan Pratt <pratt@cs.stanford.edu>*Date*: Sat, 01 Jan 2000 14:03:53 -0800*In-reply-to*: Your message of "Sat, 01 Jan 0100 11:37:15 GMT." <E124MqR-0006i7-00@owl.dpmms.cam.ac.uk>*Sender*: cat-dist@mta.ca

Thanks, Peter. One of these days I'll learn to stop sending email after midnight. Continued fractions provide equally good coalgebraic structure for both our product-with-omega functor (it's one of the examples in our paper) and Peter F.'s X v X functor. Either midnight madness or sheer forgetfulness must have possessed me to malign its applicability to the latter. While I have that excuse handy let me also repair my description (in the same message) of halving nonzero reals as right-shifting with sign extension: following the shift the second bit must then be complemented. Thus + (1/2) halves to +- (1/2 - 1/4 = 1/4) while +++ (1/2 + 1/4 + 1/8 = 7/8) halves to +-++ (1/2 - 1/4 + 1/8 + 1/16 = 7/16). In the special case of 1 as ++++... forever, +-+++... equals + (1/2), and dually for -1. Contorted fractions make an earlier appearance in Conway's On Numbers and Games (1976) (Winning Ways is 1983). I hadn't realized Norton was involved there: Conway credits several things to Norton in ONAG but I guess he must have forgotten that one. At the risk of turning this thread into a complete tangent space, yet another construction of the group of reals is as the quotient G/H of the pointwise-additive group G of bounded integer sequences by the subgroup H consisting of those sequences b of the form b_0 = a_0, b_{i+1} = a_{i+1} - 2a_i for some a in G. This definition, which avoids detouring through the rationals, resulted from my mulling over a talk at MIT by Gian-Carlo Rota in the early 1970's on representing reals as sequences of bits. I mentioned it at a recent theory lunch talk and Don Knuth mulled it over and came up with the idea of modifying the boundedness condition to allow G to be a ring thus making G/H a field (as an alternative to taking product to be the unique bilinear operation * satisfying 1*1 = 1), see Problem 10689, American Mathematical Monthly, 105(1998), p.769. I would love to know whether this construction can exploited in a coalgebraic setting. Vaughan Pratt

**Follow-Ups**:**categories: Re: Real interval halving***From:*Michael Barr <barr@barrs.org>

**References**:**categories: Real interval halving***From:*"Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>

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