Solution to a problem of FitzGerald (Q2037087)

Here, dependencies of properties of an algebra and its endomorphism monoid are considered. It is known that if the monoid of endomorphisms of an algebra has commuting idempotents, then the algebra itself should satisfy four conditions \((RI)\), \((UR)\), \((RI*)\), \((UR*)\). The open problem was: are these conditions also sufficient for an algebra to have commuting idempotents in its monoid of endomorphisms? Here, a negative answer is given. It is also shown, that for any algebra \(A\) with endomorphism monoid \(S\) the four properties for \(A\) do hold iff they also hold for \(S\) as the right \(S\)-set under multiplication. If \(S\) satisfies \((UR)\), \((RI*)\), \((UR*)\) then \(A\) also does, but for the property \((RI)\) an example of an algebra is provided whose right \(S\)-set satisfies \((RI)\), but the algebra itself may not satisfy the property.