Let X be a set and T : X → X
be a bijection. Consider the space ℳ of pseudo-Menger maps on X which induce a
compact topology on X for which T is a homeomorphism. The lattice properties of
ℳ are investigated and a bivariate nonnegative function of ℳ is defined
which possesses certain properties analogous to those of the usual conditional
entropy function defined on the space of measurable partitions of a probability
space.