Embedding Sums of Cancellative Modes into Semimodules

A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice. The paper was written within the framework of COST Action 274. Research supported by Warsaw University of Technology under grant number 504G/1120/0008/000.