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**Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?**

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**Today**at 00:41:29 »

And I'm reasonably sure now that I can describe this poset graph as a Boolean lattice, since by inspection the chains are multiplied at each part of the whole partition. There is 1, then 1 + 1, then 1 + 2 + 1, 1 + 6 + 1, . . . ,as there is always 1 lower and 1 upper chain, with the multiple chains in the interior.

The first minimal chain is just the first abstract tile, a 1-dimensional line embedded in Z

And by restricting the generators of the poset to two instead of all three, the sizes of each equivalence class are reduced to 2

One other thing that's possible if you remove or forget the identity permutation from the poset, then you have 7! x 3

Something along those lines. Also how the algorithmic aspects of finding a solution, for a given permutation, will always need to employ reduction and restriction, such that an inductive "path" exists.

Dilworth's theorem and its dual allow the decomposition of a poset into chains and antichains with a relation between them which is derived from Greene's theorem. It's quite an elegant way to analyse everything about the structure of the poset graph, but it doesn't give you the total, saturated poset because the B

The first minimal chain is just the first abstract tile, a 1-dimensional line embedded in Z

_{2}x Z_{3}. It represents the cover of the partition at (0,0), the identity permutation. This singleton can be contracted to a loop on the point (1,1). This is ok for various reasons, and the simplest reason I can cook up is that the "solution" can be defined as "less than or equal to one move" from this permutation. So an algorithm is free to halt when it's at (1,1) or (1,2).And by restricting the generators of the poset to two instead of all three, the sizes of each equivalence class are reduced to 2

^{n}, for some n. This is an example of a restriction (2 of 3 generators) which reduces the cardinality of a set. Then the dual or adjoint induction is the Boolean lattice, with a ring algebra which should then give some rules of composition for the tiles. There are 54 in total, arranged as parts of chains and antichains.One other thing that's possible if you remove or forget the identity permutation from the poset, then you have 7! x 3

^{7}- 1 total permutations, which factors to 17 x 216127. So the two prime factors also have something to say about the complexity of this problem, and the limits on packing "information" into three dimensions.Something along those lines. Also how the algorithmic aspects of finding a solution, for a given permutation, will always need to employ reduction and restriction, such that an inductive "path" exists.

Dilworth's theorem and its dual allow the decomposition of a poset into chains and antichains with a relation between them which is derived from Greene's theorem. It's quite an elegant way to analyse everything about the structure of the poset graph, but it doesn't give you the total, saturated poset because the B

_{n}lattice is a subset of the total lattice.