Arithmetic of ordered sets more general than ordinals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:28:03Z http://mathoverflow.net/feeds/question/38460 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38460/arithmetic-of-ordered-sets-more-general-than-ordinals Arithmetic of ordered sets more general than ordinals Antonio E. Porreca 2010-09-12T10:04:32Z 2010-09-13T02:14:21Z <p><b>Motivation.</b> Having read about <a href="http://dx.doi.org/10.1023/A:1021180801870" rel="nofollow">infinite time Turing machines</a> and <a href="http://en.wikipedia.org/wiki/Omega_language" rel="nofollow"><i>&omega;</i>-languages</a>, I was thinking about more general notions of languages and “computation time”. Languages over strings of length greater than <i>&omega;</i> seem reasonably easy to define, and using larger ordinals to measure time is quite standard.</p> <p>However, I’m interested in more general, non necessarily well-founded ordered sets. Using ordered sets to keep track of time seems to require suitable notions of arithmetic operations. For example, it’s easy to generalise sum and multiplication of ordinals to arbitrary totally ordered sets (though, of course, these operations are not defined by transfinite recursion); I’m not sure about exponentiation and other operations.</p> <p><b>Question.</b> Is there any good literature about arithmetic of (some classes of) non well-founded ordered sets?</p> http://mathoverflow.net/questions/38460/arithmetic-of-ordered-sets-more-general-than-ordinals/38519#38519 Answer by Yuval Filmus for Arithmetic of ordered sets more general than ordinals Yuval Filmus 2010-09-12T23:43:54Z 2010-09-12T23:43:54Z <p>I recommend Rosenstein's excellent book "Linear orderings".</p> http://mathoverflow.net/questions/38460/arithmetic-of-ordered-sets-more-general-than-ordinals/38526#38526 Answer by Aaron Sterling for Arithmetic of ordered sets more general than ordinals Aaron Sterling 2010-09-13T02:14:21Z 2010-09-13T02:14:21Z <p>Peter Aczel's book <a href="http://csli-publications.stanford.edu/site/0937073229.shtml" rel="nofollow">Non-Wellfounded Sets</a> would be my suggestion.</p>