PRODUCTS OF ULTRAFILTERS AND MAXIMAL LINKED SYSTEMS ON WIDELY UNDERSTOOD MEASURABLE SPACES

Abstract: Constructions related to products of maximal linked systems (MLSs) and MLSs on the product of widely understood measurable spaces are considered (these measurable spaces are defined as sets equipped with π-systems of their subsets; a π-system is a family closed with respect to finite intersections). We compare families of MLSs on initial spaces and MLSs on the product. Separately, we consider the case of ultrafilters. Equipping set-products with topologies, we use the box-topology and the Tychonoff product of Stone-type topologies. The properties of compaction and homeomorphism hold, respectively.


Introduction
In this investigation, properties of maximal linked systems (MLSs) and ultrafilters on widely understood measurable spaces (MSs) are considered. Every such MS is realized by equipment of a nonempty set with π-system of subsets of this set with "zero" and "unit" (the "zero" is an empty set, and the "unit" is our original set); a π-system is a family closed with respect to finite intersections. Of course, algebras, semi-algebras, topologies, and families of closed sets in topological spaces (TSs) are π-systems. An important variant of a π-system is realized by a lattice of subsets of a fixed nonempty set. A semi-algebra of sets is a π-system but, generally speaking, not a lattice.
We note that MLSs were considered in connection with the superextension and supercompactness problem, see [2,16,17,20,21]. In addition, MLSs on the lattice of closed sets in a TS were studied. The nonempty set of all MLSs of such type is equipped with Wallman-type topology. The supercompactness property was implemented.
In [5-7, 9, 10, 12], an analog of the superextension and supercompactness property for the space of MLSs on a π-system was investigated. Moreover, a Stone-type topology was also used. In addition, a bitopological space was implemented. The present study continues the above works. But here the focus is on spaces of MLSs with Stone-type topology. We consider questions related to the products of widely understood measurable spaces. In addition, representations of MLSs on the product of these MSs in terms of analogous MLSs on spaces-factors are indicated. Namely, MLSs on the product of (widely understood) MSs are limited to products of MLSs on initial spaces. This important property is complemented by a proposition of a topological nature: the properties of compaction and homeomorphism hold. In addition, the box and Tychonoff variants of topology product are considered (similar variants are used for the product of MSs). In connection with the above assumptions, we use constructions of [11,13,14].

General notions and notation
We use standard set-theoretic notation, including quantifiers and propositional connectives; ∅ stands for an empty set) and △ = for an equality by definition. A family is a set such that all its elements are sets themselves. We adopt the axiom of choice. For every objects x and y, denote by {x; y} an unordered pair of x and y: x ∈ {x; y}, y ∈ {x; y}, and (z = x) ∨ (z = y) for every z ∈ {x; y}. For every object s, denote by {s} △ = {s; s} a singleton containing s : s ∈ {s}. In addition, sets are objects. Then, for every objects x and y, the family (x, y) △ = {{x}; {x; y}} is (see [12, Ch. II, Section 2]) the ordered pair with x as the first element and y as the second. For every ordered pair h, denote by pr 1 (h) and pr 2 (h) its first and second elements, respectively; thus, h = (pr 1 (h), pr 2 (h)).  is realized. If A is a nonempty family and B is a set, then is the trace of A onto the set B. Following to [7,Section 1], if X is a nonempty family, then {∪}(X), {∩}(X), {∪} ♯ (X), and {∩} ♯ (X) stand for the families of arbitrary unions, arbitrary intersections of nonempty subfamilies of X, finite unions, and finite intersections of sets from X, respectively.

Remark 1.
In what follows, we use two types of formulas. Namely, we use expressions of type {x ∈ X| . . .} and expressions of type {f (z) : z ∈ . . .}. In function theory, the former is used for the preimage of a set; we have a formula corresponding to Zermelo-Fraenkel axiomatic (we first select a set X, for points of which some property . . . is postulated). The second expression corresponds logically to the image of a set. This difference is essential from point of view of bibliographic references to earlier publications of the author. Therefore, we use two variants of separator character: | (vertical line) in the first case and : (colon) in the second. This stipulation is important for the constructions that follow.
For sets A and B, we denote by B A (see [19, Ch. II, § 6]) the set of all mappings (functions) from A to B; values of mappings are denoted in traditionalway. If A and B are sets, f ∈ B A , and with "zero" and "unit": π[I] △ = I ∈ P ′ (P(I))| (∅ ∈ I)&(I ∈ I)&(A ∩ B ∈ I ∀A ∈ I ∀B ∈ I) . (2.1) Of course, P(I) ∈ π[I]. Consider a very useful notion of semi-algebra of sets. For L ∈ π[I], A ∈ P(I), and n ∈ N, we introduce finite partitions of A by sets of L: The family of all semi-algebras of subsets of I is defined as follows: In addition, we introduce yet another type of π-systems; this type is important in questions of interconnection between ultrafilters and MLSs. Namely, Of course, very general constructions are connected with lattices. The family of all lattices of subsets of I with "zero" and "unit" is Moreover, Thus, we have introduced open bases and subbases of the specific TS (I, τ ).
Linkedness. If J ∈ P ′ (P(I)), then we suppose that Elements of the family (2.8 and only they are linked subfamilies of J . As a corollary, is the family of all maximal linked subfamilies of J . We call every family of (2.9) an MLS (on J Finally, note that, for E ∈ J − link 0 [I], we have More detailed information on the properties of MLSs can be found in [5][6][7][9][10][11][12]. Now we introduce some constructions for a Stone-type topology. If J ∈ J , then The sets (2.11) define an open subbase. More precisely, the subbasê generates the following topology of Stone type: . (2.12) In addition, ( J − link 0 [I], T * I|J ) is a zero-dimensional T 2 -space.

Generalized Cartesian products
In this sections, we recall some constructions connected with Cartesian products and generalized Cartesian products. We note also some notions connected with family products.
If X and Y are nonempty sets, X ∈ P ′ (P(X)), and Y ∈ P ′ (P(Y )), then (X × Y is the usual product of X and Y, i.e., the set of ordered pairs); (3.1) is the simplest variant of the constructions used below. It is easy to verify the property We consider (X × Y, X {×}Y) as the product of the MSs (X, X ) and (Y, Y). Now we recall notions connected with generalized Cartesian products. If X and Y are nonempty sets and (Y x ) x∈X ∈ P ′ (Y) X , then (by the axiom of choice) In connection with (3.3), note that, for every nonempty sets X,Ỹ, andŶ and a mapping In what follows, in constructions of type (3.3) we take into account (3.4). If X and Y are nonempty sets and (Y x ) x∈X ∈ P ′ (Y) X , then We consider the family (3.5) as a box product of the families E x , x ∈ X. Here, we note the natural analogy with the base of the known box topology (see [18,Ch. 3] . As a corollary, for nonempty sets X and Y, a mapping (Y x ) x∈X ∈ P ′ (Y) X , and a mapping ( (3.6) In connection with (3.5), note that, for every nonempty sets X and Y, a mapping (Y x ) x∈X ∈ P ′ (Y) X , and a mapping ( In connection with (3.6), note that, for the above X, Y, (Y x ) x∈X , and (Y x ) x∈X , we have (3.8) Note useful particular cases of (3.7) and (3.8): for nonempty sets X and Y and mappings Using (2.7) in (3.9), we obtain two variants of topological equipment: Namely, by (3.10), we obtain the following two TSs: thus, we obtain the box TS and the Tychonoff product. Of course, topologies (3.10) are comparable. Moreover, for every nonempty sets X and Y and mappings (Y x ) x∈X ∈ P ′ (Y) X and From (3.11), the comparability of topologies (3.10) follows, since Thus, for every nonempty sets X and Y and mappings

Ultrafilters and maximal linked systems
In this section, we fix a nonempty set E and a π-system L ∈ π[E]. Recall the notions of filter and ultrafilter on this π-system. So, is the set of all filters on L. Hence (see [7, Section 2]), We recall that F * 0 (L) = ∅ (this is a simplest corollary of the Zorn Lemma). If L ∈ L, then In connection with (4.3), note that (F * 0 (L), T * L [E]) is a zero-dimensional T 2 -space, see [3]. Thus, In what follows, we use the inclusion F * 0 (L) ⊂ L − link 0 [E], see [8, (3.2)]. Now, we recall one general property (see [8, (4.2)]): In this connection, note that (see [8, (3.12)]), in the general case of L, we have In connection with (4.4), we note [8, (4.3)] where supercompactness conditions for a topology of Wallman type were considered. Moreover, in the general case of L ∈ π[E], we have the following representation [8, (4.1)]: Therefore, we obtain the following useful equality: It is easily to verify that
Example 1. Assume that X = Y = 1, 3; thus, X = Y is a three-element set: 1 ∈ X, 2 ∈ X, and 3 ∈ X. Suppose that X = P(X) and Y = P(Y ); of course, X = Y. Now, we introduce the linked family E by the rule X × {2} ∈ E, {2} × Y ∈ E, {(2, 2)} ∈ E, and the family E does not contain any other sets. So, E is a specific three-element family. Of course, However, Using (5.2), we find that X ∈ A and Y ∈ B. Then, X × Y ∈ A{×}B. But X × Y / ∈ E. The obtained contradiction proves the required property: E does not have a rectangular structure.
and the following property: In addition (see (2.10)), X ∈ U 1 and Y ∈ U 2 . Consider an MLS U 1 . For this, we fix M ∈ U 1 , N ∈ U 1 , and T ∈ U 1 . Then, by the choice of (see (2.10)). From (5.13) and (5.14), we obtain M ∩ N ∩ T = ∅. Since the choice of M, N, and T was arbitrary, the inclusion Theorem 1. The following equality holds: The proof reduces to immediate combination of Propositions 2 and 3. Finally, we note an important property of topological character (see [13,Theorem 5.1]). We recall that, by (3.1) and (3.2), then, by (2.7), the natural topology of the product of Stone-type TSs is realized. Moreover, the following Stone-type topology is defined: Then, by [13, Theorem 5.1], the mapping is a homeomorphism from the TS Note that, by (4.7), we have Moreover, using (4.5), we obtain So, ultrafilters of π-system X {×}Y form a closed subspace of TSs homeomorphic to (5.16). Theorem 1 reveals the structure of this subspace.
6. Infinite products of maximal linked systems, 1 Unless otherwise stated, in what follows, nonempty sets X and E and a mapping (E x ) x∈X ∈ P ′ (E) X are fixed (for x ∈ X, we denote by E x a nonempty subset of E). Define the set E (hereinafter, the axiom of choice is used). Finally, we fix We obtain (see (6.2)) the following two variants of π-systems: (we use [8, (6.4)-(6.5)]); in connection with (6.3)-(6.5), we recall (3.6)-(3.8). So, we have two comparable π-systems on E.
Now, we note one simple property: Moreover, we note that The property (6.7) assumes a natural development; now, we note only that By (6.6), an obvious corollary is realized; namely, Using (6.9), we define a mapping P : In addition, by (6.9), we have Now, note the following obvious inclusions: Of course, (6.13) defines the corresponding projection mapping. From (6.11) and (6.13), for χ ∈ X and (Σ x ) x∈X ∈ x∈X P ′ (E x ), we obtain P χ x∈X Σ x = Σ χ . (6.14) From (6.12) and (6.14), we, in particular, obtain Using the notion of the set image, we suppose that ∀H ∈ P(( x∈X P(E x )) \ {∅}) ∀χ ∈ X Then, the following obvious property holds: if H ∈ P(( x∈X L x ) \ {∅}) and χ ∈ X, then We can use a natural combination of (5.3) and (6.16): a linked system can be used as H. In addition, by [13, Proposition 3.2], we have As a corollary, for E ∈ x∈X L x − link [E], we obtain the mapping This proposition corresponds to [13, Proposition 3.1]. To prove Proposition 4, it suffices to use (6.7) (and the axiom of choice). From (6.17) and Proposition 4, we obtain Then, (6.17) is supplemented by the following statement: Moreover, by [13, Proposition 3.6], we obtain the following property: By (6.19) and (6.20), the following basic statement (see [13,Theorem 3.1]) holds: In (6.21), we have a natural analog of (5.8). In connection with (6.21), we note that Then, by (6.21) and (6.22), we obtain Recall (4.6) and (6.4). Then, by (4.6) and (6.23), we have Let A ∈ x∈X U x , B ∈ x∈X U x , and let C ∈ x∈X U x . Then, by (3.7), for some we obtain the following equalities: From (6.22), for x ∈ X, we obtain the inclusions A x ∈ P(E), B x ∈ P(E), and C x ∈ P(E). Then, by (6.8) and (6.26) In addition, for x ∈ X, we obtain A x ∈ U x , B x ∈ U x , and C x ∈ U x ; then, by (4.6) and (6.24) Using (6.27) (and the axiom of choice), we obtain A ∩ B ∩ C = ∅. Since the choice of A, B, and C was arbitrary, it is established that the premise of implication (6.25) is true. So, we obtain the required property By formula (4.6), we get Therefore, by formula (4.6), we get Choose arbitrary A ∈ E χ , B ∈ E χ , and C ∈ E χ . By (5.3), A ∈ P ′ (E), B ∈ P ′ (E), and C ∈ P ′ (E). Now, we introduce (Ã x ) x∈X ∈ P ′ (E) X by the rule Similarly, we introduce (B x ) x∈X ∈ P ′ (E) X by the rule Finally, define (C x ) x∈X ∈ P ′ (E) X by the rule Then, by (6.8), we obtain the following obvious equality: By the choice of (E x ) x∈X , we obtain (see (3.7)) x∈XÃ As a corollary, by (6.28), we have the following important statement: Then, from (6.30), we obtainÃ x ∩B x ∩C x = ∅ for x ∈ X. In particular, A χ ∩B χ ∩C χ = ∅. As a corollary, A ∩ B ∩ C = ∅. Since the choice of A, B, and C was arbitrary, the following property holds: From (6.29), we obtain E χ ∈ F * 0 (L χ ). Since the choice of χ was arbitrary, As a corollary, by the choice of (E x ) x∈X , we obtain Theorem 2. The following equality is true: The proof immediately follows from Propositions 5 and 6. Returning to (6.21), we note that is a surjection. Moreover (see (2.11)), by [14,Proposition 4.3], for (L x ) x∈X ∈ x∈X L x , we have Moreover, the following set-product is defined: In addition (see Section 2), ) for x ∈ X. Then, by (3.5), we have thus, the box product of the familiesĈ * 0 [E x ; L x ], x ∈ X, is defined. Moreover, we have the propertŷ From (6.32), we obtain the following statement: Now, we recall that (see (2.12)), for x ∈ X, By (3.9), is used as an open base for the corresponding box topology: Moreover, by (6.34), we obtain On the other hand, by (2.12), the following inclusion holds: Therefore, from (6.33) and (6.35), we find that f is a continuous mapping in the sense of topologies condenses on the following space of Stone type: In addition, by (4.7), we obtain Theorem 2 reveals the structure of the set F * 0 ( x∈X L x ). By (4.5), we have thus, ultrafilters of the π-system x∈X L x form a closed subspace of the space (6.38).
Theorem 3. The following equality holds: The proof immediately follows from Propositions 7 and 8. In connection with Theorem 3, we recall constructions of [4].
Following to [14], we introduce the following natural mapping: The properties of g (see (7.25), (7.26)) were considered in [14]. Now we will restrict ourselves to listing them. Note that (in (7.27), we use (3.6) and take into account that, for x ∈ X, Now, we recall (6.34). As a corollary, the following π-system is defined: (7.29) we use (3.6) and (3.9). By means of (2.7), (3.10), and (7.29), the topology is defined. From (6.34) and (7.30), we obtain Therefore, by (7.28) and (7.31), we have the following property: Using (6.36) and (7.32), we obtain the following important property: g (7.25) is a continuous mapping in the sense of TS (in this connection, we recall (7.6)). In addition, we recall the following useful statement [14, Proposition 6.5]: By means of this property, the following important statement was established in [14, Proposition 7.1]: g is an open mapping in the sense of TS (7.33). So, we obtain the following basic statement (see [14,Theorem 7.1]).
Theorem 4. The mapping g (7.25) is a homeomorphism from the TS From (4.7), we obtain Theorem 3 reveals the structure of the set F * 0 ( x∈X L x ). By (4.5), we have Thus, ultrafilters of our π-system x∈X L x form a closed subspace of the second TS in (7.33).

Some corollaries for ultrafilter spaces
In this section, we consider some statements related to products of spaces with topologies of type (4.3). But, at first, we note general properties connected with subspaces of TSs.
For every TS (X, τ ), X = ∅, and (Y, ϑ), Y = ∅, denote by C(X, τ, Y, ϑ) the set of all mappings from Y X continuous with respect to the topologies τ and ϑ. Similarly, for nonempty sets X and Y, let be the set of all bijections from X onto Y; finally, for τ 1 ∈ (top)[X] and τ 2 ∈ (top)[Y], let be the set of all condensations from (X, τ 1 ) onto (Y, τ 2 ). We note yet another important notion: for every TS (X, τ 1 ), X = ∅, and (Y, be the set of all open mappings from (X, τ 1 ) in (Y, τ 2 ). Then is the set (possibly empty) of all homeomorphisms from (X, τ 1 ) onto (Y, τ 2 ). Now, we note several simple general properties.
Immediate combination of (1) and (2) implies the following properties. Now, we note some statements on the structure of a subspace of the product of TSs. If (X, τ 1 ), X = ∅, and (Y, τ 2 ), Y = ∅, are two TS, then, similarly to Section 5, in what follows, we suppose that Note that (3.6) and (8.2) should be distinguished; in (8.2), we consider a topology. Then, using [15,Proposition 2.3.2], for every TS (X, τ 1 ), X = ∅, and (Y, τ 2 ), Y = ∅, and sets A ∈ P ′ (X) and of course, we keep in mind that, in the case under consideration, In (8.4), we have an analogy with [15,Proposition 2.3.2] (an obvious verification of (8.4) we omit). Finally, for every nonempty sets X and Y, mappings Now, we consider some topological properties for products of ultrafilter spaces. We begin with the simplest case.
The case of box topology on the product of ultrafilter spaces. In this and subsequent subsections, we use nonempty sets X and E and the mapping (E x ) x∈X ∈ P ′ (E) X defined in Section 6. Moreover, we follow (6.1) for the set E. In what follows, we fix (L x ) x∈X (6.2). Then, by (4.3), we have In addition, Using (4.5), we obtain From (3.10) and (8.13), the following property is extracted: We recall that, by Proposition 5, the mapping is defined correctly. By (6.31), this mapping (8.16) is a restriction of (6.31) to the set x∈X F * 0 (L x ). For brevity, we denote the mapping (8.16) by w. By (6.22), we have and Moreover, we recall that, by (6.31) and Theorem 2, Now, we use (3) with the following specific definitions: By (8.19), the following inclusion holds: Now, we use (8.4) with the following specific definitions: Using (8.23), we also suppose that In this connection (see (8.23) and (8.24)), we recall that, by (2.8) and (2.9), the following chain of inclusions holds: Therefore (see (8.23)), This corresponds to the conditions for (8.4). Then, from (8.4), (8.23), and (8.24), we have The case of generalized Cartesian product of ultrafilter spaces. We follow the previous subsection (see also Sections 6 and 7), using X, E, (E x ) x∈X , E, and (L x ) x∈X . Of course, we use (8.13)-(8.14). Then, by (3.10) and (8.13), we have From Proposition 7, we conclude that is a restriction of the mapping g (7.25) to the set x∈X F * 0 (L x ). We denote this mapping (8.28) by r for brevity; so,

Conclusion
In this paper, some questions related to the structure of ultrafilters and MLSs on products of widely understood MSs were considered. In this connection, two basic directions were developed: the direction connected with representations for ultrafilter and MLSs on the products of MSs (set-theoretical direction) and (topological) direction connected with topological relations between TSs of Stone type arising under consideration of topology products (in the box and Cartesian variants) and topologies on the sets of ultrafilters and MLSs for the product of the corresponding measurable structures. In the first direction, the following property is established: ultrafilters and MLSs on products of MSs are exhausted by products of ultrafilters and MLSs, respectively. In the second direction, important properties of homeomorphism and compaction were obtained. In addition, the compaction property is established for the box products of TSs. In the case of the generalized Cartesian product, the homeomorphism property holds. This comparison shows the better character of Tychonoff's product of TSs compared to box TSs.