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

Abstract 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.