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New shift to clone theory.

Item posted by Daniel Wyckoff <deluger> on Sun 04 Nov 2018 04:37:00 AM UTC.

Now that I realized that really the Sikorski Extension and various corollaries and half-starts must and can be managed ideally (under the automorphism model discussed on the home page), and discussed this in  <a> href ="http://web.cvs.savannah.nongnu.org/viewvc/bool2/bool2/theory/goodaut.tex">/a>, I will change much of the direction of future development.  I've placed cleaning up bool2 metaphysically as a higher priority than adding haphazard automorphisms, and luckily I came to the intuitive insight that much if not all of these "magical automorphisms" as described in the documentation can be reduced to permutations of atoms of inducing constructs parameterized by elementary functions -- closed under composition (the latter to ensure compatiblity amongst Boolean operators and permutations thereof) -- this being the definition of clone, quite fortunately!  So, the clones and their automorphism group seem to determine the behavior of the set of all representations of S_n that permute relations -- on the "birelational" space discussed in the documentation.  Luckily this automorphism tower gets so computationally and conceptually unmnageable on even small finite sets, this automorphism tower of clean-up will probably not need to continue.  When it comes down to it, no one asked me directly to program bool2, unless you believe in superstition, but if that's the case, then I must believe in this strong superstition which if false will still point in the direction to the right answer upon further reflection.  So, now I will slowly formalize clones -- I have Googled far and wide and searched the standard Coq User Contributions on Inria, and not found a single entry in clone theory, so will begin.  This will be more challening and more of a need than a desire, out of Jewish guilt!  The idea that this program causes office air or some such in some signifcant sense is enough for me to superstitiously stick with a lot of the theorems in "Algebras and Orders" by Rosenberg and Sabidussi -- but I will consider many entries in the Springer monograph, too.  The idea, once more, is to induce automorphisms in the "relational automorphism group" (whether or not it's technically a BAO automorphism group in the conventional sense I would have to consider) -  bringing to mind an automorphism tower that on the second iteration will once again not necessitate a third, by virtue of huge computations and unknown theorems.  Intuitively the clone2 offshot will ensure a wholeness by manifesting more logic (the full set of clones which determine a full set of "relational permutations"), and more mystery (their unwiledy nature).   Happy Hacking!

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