try the three element partial order monotonic maps
what constraint can you put on two sides of the math
anything else/seriously different doesn't generalize enough
map needs to always be onto/equivariant, - constraint on either side, constraint on space monoid is acting on, either side of the map
why can't i just make the map monotonic pushes back to square one
if there is a partial ordering on either side, and that nu is equivariant, then nu is monotonic.
if we can't prove this, we don't have the equivariance at the group/semi-group level, so need to jump to categories
structure preserving map is equivariant map
if we impose constraint that is equivariant map, need to recover that if equivariant map, that means map is monotonoic for ordered states
can't prove that map equivariant to monoid actions is monotonic
F having structure means is a category
if there is a partial ordering on either side, and that nu is equivariant, then nu is monotonic. if we can't prove this, we don't have the equivariance at the group/semi-group level, so need to jump to categories structure preserving map is equivariant map if we impose constraint that is equivariant map, need to recover that if equivariant map, that means map is monotonoic for ordered states can't prove that map equivariant to monoid actions is monotonic F having structure means is a category