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modes (idempotent and entropic algebras); cancellative modes; sums of algebras; embeddings; semimodules over semirings; idempotent subreducts of 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.
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