On projections of the tails of a power

Abstract

Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and

the equivalence relation on

U κ \mathfrak{U}^{\kappa}

of eventual equality. From mild assumptions on 𝜅, we give general constructions of

E ∈ End ( U κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathfrak{U}^{\kappa}/{\sim})

satisfying

E ∘ E = E \mathcal{E}\circ\mathcal{E}=\mathcal{E}

which do not descend from

Δ ∈ End ⁡ ( U κ ) \Delta\in\operatorname{End}(\mathfrak{U}^{\kappa})

having small strong supports. As an application, there exists an

E ∈ End ( Z κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathbb{Z}^{\kappa}/{\sim})

which does not come from a

Δ ∈ End ⁡ ( Z κ ) \Delta\in\operatorname{End}(\mathbb{Z}^{\kappa})

.