Cancellation in entropic algebras

We describe the equational theory of the class of cancellative entropic algebras of a fixed type. We prove that a cancellative entropic algebra embeds into an entropic polyquasigroup, a natural generalization of a quasigroup. In fact our results are even more general and some corollaries hold also for non-entropic algebras. For instance an algebra with a binary cancellative term operation, which is a homomorphism, is quasi-affine. This gives a strengthening of K. Kearnes’ theorem. Our results generalize theorems obtained earlier by M. Sholander and by J. Ježek and T. Kepka in the case of groupoids. The work on this paper was conducted within the framework of INTAS project no. 03 51 4110 “Universal algebra and lattice theory”. The author was also supported by the Statutory Grant of Warsaw University of Technology no. 504G11200013000.