Stochastic tensors and approximate symmetry

Triply stochastic cubic tensors, or sharply transitive sets of doubly stochastic matrices, are decompositions of the all-ones matrix as the sum of an ordered set of bistochastic matrices. They combine to yield so-called weak approximate quasigroups and Latin squares. Approximate symmetry is implemented by the stochastic matrix actions of quasigroups on homogeneous spaces, thereby extending the concept of exact symmetry as implemented by permutation matrix actions of groups on coset spaces. Now approximate quasigroups and Latin squares are described as being strong if they occur within quasigroup actions. We study these weak and strong objects, in particular examining the location of the latter within the polytope of triply stochastic cubic tensors. We also establish the rudiments of an algebraic structure theory for approximate quasigroups. Upon relaxation from probability distributions to their supports, approximate quasigroups furnish non-associative analogues of (set-theoretical) hypergroups.