Maximal linked systems on families of measurable rectangles

Linked and maximal linked systems (MLS) on $\pi$-systems of measurable (in the wide sense) rectangles are considered ($\pi$-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with $\pi$-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or $\sigma$-algebra of sets. In the general case, the family of measurable rectangles itself forms a  $\pi$-system of set-product (the measurability is identified with belonging to a  $\pi$- system) which allows to consider MLS on a given $\pi$-system (of measurable rectangles). The following principal property is established: for all considered variants of $\pi$-system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.