Abstract:Medial modes, a natural generalization of normal bands, were investigated by P\l onka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by P\"oschel and Reichel. In this paper we show that each medial mode embeds as a subreduct into a semimodule over a certain ring, and that a similar theorem holds for each Lallement sum of cancellative modes over a medial mode. Similar results are obtained for rectangular algebras. The paper generalizes earlier results of A. Romanowska, J.D.H. Smith and A. Zamojska-Dzienio.
Keywords: modes (idempotent and entropic algebras), cancellative modes, sums of algebras, embeddings, semimodules over semirings, idempotent subreducts of semimodules
AMS Subject Classification: Primary: 08A05; Secondary: 03C05, 08C15