On Mode Reducts of Semimodules

Modes are idempotent and entropic algebras. More precisely, an algebra (A,Ω) of type τ : Ω −→ Z is called a mode if it is idempotent and entropic, i.e. each singleton in A is a subalgebra and each operation ω ∈ Ω is actually a homomorphism from an appropriate power of the algebra. Both properties can also be expressed by the following identities: (I) ∀ω ∈ Ω, x . . . xω = x (E) ∀ω, φ ∈ Ω, with m-ary ω and n-ary φ,