Fixed point structures on a set-mapping pair and cartesian product

In this paper we study the following problem (Problem 4.2 in, I.A. Rus, Sets with structure, mappings and fixed point property: fixed point structures, Fixed Point Theory 23, No. 2 (2022), 689-706):

Let (𝒰, 𝒮, M) be a fixed point structure on (𝒰, M). We suppose that (𝒰, M) is with cartesian product. In which conditions, we have that:

X,Y∈𝒮⇒X×Y∈𝒮?

Some results in terms of exponential with fixed point property are also given. The basic results are illustrated by examples.