On projections of the tails of a power

Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∼</m:mo> </m:math> ~ the equivalence relation on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> </m:math> Uκ of eventual equality. From mild assumptions on 𝜅, we give general constructions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo rspace="0em">/</m:mo> <m:mo lspace="0em" rspace="0em">∼</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> E(Uκ/{~}) satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo lspace="0.222em" rspace="0.222em">∘</m:mo> <m:mi mathvariant="script">E</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant="script">E</m:mi> </m:mrow> </m:math> E=E which do not descend from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>End</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="fraktur">U</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> Δ(Uκ) having small strong supports. As an application, there exists an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">E</m:mi> <m:mo>∈</m:mo> <m:mi>End</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo rspace="0em">/</m:mo> <m:mo lspace="0em" rspace="0em">∼</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> E(Zκ/{~}) which does not come from a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>End</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>κ</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> Δ(Zκ) .