Medial modes, a natural generalization of normal bands, were investigated by Płonka. Rectangular algebras, a generalization of rectangular bands (diagonal modes) were investigated by Pöschel 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...
In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class...
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.
Download Results (CSV)