Calculating with sets

Three sets of sets with integer indexed elements,x,y,z .

Define resolution. as the focal curve, x/y.  The number of ways one can take y for the number of time x.

Confine our set to those for which:

y/xz  are the number of partitions the set of y elements can make on txz paths, all paths considered.

Call that the resolution of the y elements over xz.  If we specify the total resolution is one Then we cn derive the Markov and the nyquist conditions together. But dimension m allows communtativity across a line of symmetry, and need be included.

The inverse of resolution is number of partitions of x over yz.  A Huffman encoder is matching resolution to a known set, in essence.  

But we can count dimensionality which allows commutatitivty among paths across a symmetry.

Counting x,y,z in increments is an artificial but useful toot. It enforces independence of arrivals. One can use a bit of set calculus, showing on partition resolution is a continued fraction of the previous. We are counting around a closed path.

Closed path and total resolution equal one are the same. The artificial counter counts out all the paths at some resolution, and that count is constant.  The counter x,y,z are constrained to solutions that have minimums in their total error.