Entropicity and generalized entropic property in idempotent n-semigroups

We investigate entropicity and the generalized entropic property in n-semigroups. These two properties are not equivalent for n-semigroups in general, but we show that there are certain classes of idempotent n-semigroups for which entropicity and the generalized entropic property are equivalent, such as Mal’cev n-semigroups and idempotent n-semigroups derived from binary semigroups. Moreover, we present an alternative description of entropicity in the variety of semigroups satisfying the identity xⁿ≈x for some n≥2. We also obtain a simple representation of the free ternary Mal’cev semigroup on two generators.