The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory

A category theory constitutes a convenient conceptual apparatus to organize the worlds of mathematical entities. The concept of fuzzy natural transformation as an abstract mapping on functors is one of the essential concepts of this theory. If we admit a piece of non-commutativity in its definition diagrams, then the ‘upward’ and the ‘downward’ diagram parts generate different result sets. In this way, we can introduce the concept of fuzzy natural transformation. We deal with the multi-fuzzy natural transformation if such a transformation is based on a multi-diagram. This paper aims to describe the multi-fuzzy natural transformation situation when the symmetric difference of the result is finite. It allows us to organize the whole spectrum of the result sets in a unique quotient algebra <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ </tex-math></inline-formula>. Different algebraic properties of this structure will be explored, and a piece of classical encoding theory will be reconstructed in the environment determined by this algebra. In particular, the concepts of the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-multi similarity, the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-similarity balls, and the abstract <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-similarity balls are introduced as some generalization of the idea of Hamming’s distances and Hamming’s balls. Finally, it is shown how to automate some verification processes in this context using an R-based programming environment.


I. INTRODUCTION
A category theory forms a general algebra-based conceptual apparatus for grasping the dynamic nature of different mathematical structures and relations between these structures. The founding idea of category theory stems from S. Eilenberg and S. McLane's groundbreaking idea from [1]- [3] to comprehend the processes of preserving mathematical structure. The same dynamic way of thinking found its reflection in the majority of younger positions devoted to category theory, such as [3]- [6]. Consequently, this theory may pretend to play a role of a new paradigm in the foundations of formal sciences. This postulate is comprehensive in light of the increasing tendency to grasp many set-theoretic operations on sets, collections, structures in a more dynamic and mappingbased depiction, such as embeddings or morphisms.
From another and no less sophisticated perspective -category theory might be viewed as an art of clever manipulations of mathematical entities in terms of arrows and commutative diagrams. Meanwhile, the idea of diagram commutativity The associate editor coordinating the review of this manuscript and approving it for publication was Yilun Shang. plays a fundamental role in different areas of formal reasoning. Indeed, the commutative diagrams constitute a reasoning device that clarifies the relationship between certain processes, problems, or mappings. Often, the diagram commutativity means that we can express the same result alternatively in two (or more) ways. One can indicate several examples of such commutative diagrams to illustrate the thesis.
• Having defined an initial problem instance (i.e., a differential equation), we can adopt a Fourier transform to it to obtain its modified instance. If the transformed problem is easier to be solved, we can return from its transformed solution to the initial instance solution via inverse Fourier transform.
• Having defined a relational semantics (given by a Kripke frame-based model and its ultrafilter extension) for a given modal logic system Th, we can exploit the idea of diagram commutativity to elaborate its corresponding algebraic semantics (in terms of modal algebra A and its dual frame A + (resp).) as depicted in Fig. 1. -due to the original Stone's ideas from [7]. The mutual relationships between Kripke frame F , its dual frame F + = ueF , its corresponding modal algebra A -denoted as F +and the dual algebra for the modal algebra A + . The semantic sense and properties of the diagram may be found in [8]. We omit explaining the diagram details as redundant from the perspective of further analysis.
As already mentioned -the idea of diagram commutativity manifests itself in many ways in category theory. One can venture to state that preserving diagram commutativity plays a role of a fundamental methodological principle of this theory and a necessary condition of existence of various categorial entities, such as products, co-products, etc. The same idea of diagram commutativity finds its transparent manifestation in the main reasoning line in category theory. It runs from categories by different types of functors (the covariants or the contravariants ones) towards the so-called natural transformations. The last ones form the most sophisticated and abstract mappings operating between functors. Its general form may be predicted employing the so-called Yoneda's lemma -initially elaborated in [9], [10]. 1 Meanwhile, the existence of a natural transformation between two given functors exactly means that the appropriate diagram for them commutes. This property is usually expressed in terms of diagram commutativity. It may doubt how the situation changes if we admit a fuzzy commutative diagram.
Independently of the category theory-oscillated research, an increasing development of algebraic and set-theoretic research on the so-called P(ω)/fin, its founding role in the advanced theory of Boolean algebras may be observed. This unique quotient algebraic structure enables of introducing a quite serious portion of set-theoretic considerations around forcing theory -as described in the seminal Kunen's monograph in [11] or in [12] to 'regular' algebraic structures as Boolean algebras. The last ones were broadly described in this perspective in Koppelberg's monograph [13], and -in a more classical pre-forcing perspective -several years earlier by R. Sikorski in [14], [15]. Indeed, this purely set-theoretic and extremely abstract algebraic nature P(ω)/fin -as a key concept of the so-called gaps and towers theorystrongly affected the character and a direction of research on them. It practically assigned this notion to the 'static,' set-theoretic paradigm of thinking in the foundation of for- 1 Yoneda's lemma introduces another type of algebraic duality called Yoneda's duality. In the author's opinion -it constitutes a convenient bridgehead to comprehend more sophisticated Grothendieck's duality, which lives in the worlds of functors between ringed spaces and modules. Their characterization and their deep mutual relation exceed the thematic scope of the paper. mal sciences. In addition, a natural conceptual hull of the concept determined by set-theoretic axioms (e.g., GCH), the so-called Haussdorf gaps or Cech-Stone's compactification -as described, e.g., in [16], [17] -makes this algebraic structure hardly elusive from the operational perspective and a dynamic paradigm of category theory.
Admittedly, research on P(ω)/fin shows a high affinity with combinatorics, but it refers to infinity combinatorics. As a result, a majority of classical results of finite combinatorics (such as partitioning properties of finite sets, Ramsey's theory) loses its natural operational, practical dimension and classical combinatorial sense 2 -even if some results are extrapolable for infinite sets. It may arise a question how one can restore or extract some connections between the theory of P(ω)/fin-algebras and finite combinatorics. Although these open questions might potentially be a sufficient research motivation, it remains a methodologically fundamental problem why to combine fuzzy category theory with the theory of P(ω)/fin-algebras at all?
Indeed, the author of the paper decides to do it expecting that • the appropriate P(ω)/fin-algebras -already discussed in set-theoretic works of Hausdorff [18], [19], of Parovicenko [20] and of Shelah [12] -help to organize all sets created from fuzzy natural transformation diagrams -(not only the pairs of sets ordered by inclusion relation), and • the algebras may be immersed in such an environment, which shows a deep affinity to the classical combinatorial encoding theory and allows us to extrapolate the combinatorial idea of a Hamming distance -as depicted in [21] -and Hamming balls for arbitrary sets. It seems that there is a couple of premises for such optimism. At first, P(ω)/fin shows some inherent relationships with the class of finite sets -independently of a leading research tendency to immerse this structure in cardinal arithmetic and infinite set theory. 3 Secondly, the same class of finite setsconnected to P(ω)/fin -is a key to mutually connecting the sets, which have (at most) finite intersection.
Meanwhile, the idea of a reasonable organization of relative sets -created at each stage of the fuzzy natural transformation diagrams -was developed in [22], [23] in a relative idealistic way. Indeed, it was idealistically assumed in [22], [23] that either the result of the 'upward' multi-composition of mappings, say A, is included in the result of the 'downward' multi-composition of mappings, say B or -conversely. Meanwhile, the compositions' results may differ in a general case or even have an empty intersection. Last but not least -there is another, more general reason to develop fuzzy category theory as considered in the paper. Namely, even in the most recent papers in fuzzy category theory -such as [24]- [26] -it dominates more a tendency to elaborate the category theory-based approaches to fuzzy sets than to consider fuzzy sets inside categorial entities.
All these lacks correspond well with the next premise to combine the theory of P(ω)/fin with fuzzy theory category: the following particular questions demand a solving (and they seem to be solvable thanks to the machinery of the P(ω)/finalgebra theory): 1) How to define the multi-fuzzy natural transformation if the 'upward' composition mapping is essentially different from the 'downward' composition mapping than relative sets are finite? 2) What can we state about the class C of all pairs (A, B) such that their symmetric difference is finite (i.e.,, A B ∈ fin) 4 ? 3) Can it be ordered in some way? 4) Which combinatorial properties does the class C have? Finally, P(ω)/fin 5 forms a convenient conceptual frame to explore combinatorial properties of the class of relative sets -obtained at each construction stage of the multi-natural transformation diagram. Indeed, it seems that P(ω)/fin might be a convenient bridgehead to construct some unique similarities balls and to reconstruct in this way a piece of classical encoding theory in this new algebraic environment.
Last but not least -one thing requires a piece of justification. Namely, we decided to explore the R language's programming-wise machinery as operational support in some operationally workable fragments of our analysis instead of Haskell -ordinary exploited in category theory contexts. This solution has been mainly dictated by further data analysis-sensitive automatizing the processes and different verification procedures discussed in the paper. R is a natural language for such attempts.

2) THE PAPER OBJECTIVES
Due to these motivation factors -the general objectives of the paper are: • to propose a realistic (less idealistic) approach to the multi-fuzzy natural transformation itself and to the organization of the spectrum of relative sets by the appropriate equivalence relation between these sets, • to organize the relative sets into the so-called composition algebra P Comp (ω) for relative sets -as a quotient algebra with formal properties of P(ω)/fin, • to exploit formal properties of P Comp (ω) to reconstruct a piece of classical encoding theory in this concep- 4 We can announce here that we intend to consider the class C of all such pairs as a subclass of a unique algebra P(ω)/fin (of subsets of ω over an ideal of finite sets). Will the concept be explained in detail later. 5 Pedantically, we will consider its unique subalgebra constructed from relative sets obtained at each stage of the multi-fuzzy natural transformation construction.
tual environment in terms of k-multi similarity, k-multi abstract similarity, k-multi similar balls, and k-multi abstract balls, • to extract some combinatorial properties of P Comp (ω) with an algorithmic anti-chain partitioning of this structure as a special focus area, • to propose a method to automate some procedures in this area (such as verification whether a given equivalent class belongs to a given k-multi abstract ball). These general paper objectives will be materialized by performing some sub-objectives, such as: • establishing the type of orders (partial, linear), which is created by P(ω)/fin or • establishing the maximal chain cardinality in this algebra. Nevertheless, the paper's objectives might be measured differently, considering that the paper analysis combines two mutually different paradigms in reasoning about the foundation of formal and technical sciences: the set-theoretic and the categorial one. In this perspective, the paper is focused on creating a methodological sound, synergy approach to fuzzy natural transformation in its operational depiction. From this point of view, the paper has one extended goal only.
Last but not least, there also exists a more practical perspective. From this perspective -the paper analysis may be viewed as elaborating a theoretic tissue to solve an initial computational problem being a specification of the above questions 2), 3) and 4). The exact formulation of the problem will be given in Section II in terms of the algebraic apparatus presented in this section.

3) THE PAPER ORGANIZATION
The rest of the paper is organized as follows. The conceptual framework of the paper analysis is given in Section II. Section III describes the idea of the multi-fuzzy natural transformation in a more and less idealistic perspective. In Section IV, it is shown how the class of relative sets may be algebraically organized employing the formal apparatus of the P(ω)/fin theory. In particular, the so-called composition algebra P Comp (ω)/fin is introduced here. Finally, a piece of combinatorics is introduced in terms of k-multisimilarity, k-multi abstract similarity, k-multi-similar balls, and k-multi-similar abstract balls. Section V contains solving the leading problems. Section VI describes state of the art, but Section VII contains conclusions of the previous analysis and closing remarks.

II. THE CONCEPTUAL FRAMEWORK OF FURTHER ANALYSIS AND THE LEADING PROBLEM FORMULATION
In this part, the conceptual framework of the analysis is put forward, and the leading problem is formulated. 6 The definitions of the lattice, distributive lattice, Boolean algebra, and its extensions may be found in each handbook of lattice theory and universal algebra. (See: [13].)

A. CONCEPTUAL FRAMEWORK OF FURTHER ANALYSIS
The notion of the category stems from the algebraic concept of the group of transformations. The term 'transformation' itself is exploited here as synonymous with the concept of mapping.
Definition 1: A category K is a triple consisting of: 1) a class O K of objects; 2) a class Morp K of morphisms (also called maps or arrows) between objects from O K ; 3) a binary operation • : (Morp K ) 2 → Morp K called composition of morphisms, which satisfy the following conditions: • If α, β ∈ Morp K , then β • α ∈ Morp K , • Identity: There is a unique neutral element in Morp K (the so-called identity morphism id), i.e.,, for each α ∈ Morp K , it holds: We sometimes consider categories as the pairs of the form: where O K is a class of objects (usually denoted by X , Y , . . . , A, B, . . . , etc.) and Morp K is a class of arrows between these objects. We usually decide for simplification provided that an arrow composition operation is defined to satisfy the conditions from point 3) of Definition 1.  • F(id X ) = id F(X ) , for each object X of K , Let us now assume that categories K and L, and functors F, G from K → L are given (see: Fig.2). Independently of their nature, one could establish a new and more abstract mapping from F into G to be so-called a natural transformation between F and G. Its rigorous definition is as follows.
Definition 4: Let C and D be two categories, and F and G be functors between them. A natural transformation η : F → G is a family mappings, which satisfy the following two conditions: a) to each object X ∈ C, a mapping η X : Fig. 2., the natural transformation between functors F and G is represented by the pair of η-components, i.e.,, by (η 0 , η 1 ).

Example 4: In
Let us repeat the following algebraic definitions at the end of this subsection.
Definition 5: A partial order is any binary relation, which is reflexive, anti-symmetric, and transitive. Definition 6: Each partial order P, in which there exists supremum and infimum for each pair of elements of P, is said to be a lattice.
Definition 7: Each lattice as a structure (A, ∨, ∧) satisfying the distributivity laws: is said to be a distributive lattice. Definition 8: Let a lattice A has the minimal element (0) and the maximal element (1). For any element a in A, if there exists an element such that a ∨ x a = 1 and a ∧ x a = 0 (a complement of a) x a , then the lattice is a complemented lattice.
Definition 9: Each distributive and complemented lattice is said to be a Boolean algebra.
The power set 2 X of a given set X , consisting of all subsets of X with inclusion relation, forms a standard example of Boolean algebra. X may be here any set: empty, finite, infinite, or even uncountable.
We now define a unique and important Boolean algebra P(ω)/fin defined as a quotient structure. Let assume that fin(X ) denotes a family of all finite subsets of X . We will write fin for X = ω (a linearly ordered set of natural numbers).

Definition 10 (P(ω)/fin): The algebra P(ω)/fin is a quotient set consisting of all equivalence classes [A]
, for A ⊆ ω, determined by the following equivalence relation ∼ fin :

B. THE LEADING PROBLEM FORMULATION
Let us consider a fuzzy natural transformation multi-diagram (as in Fig. 4), i.e.,, where the commutativity condition is violated, and different relative sets arise. Assuming that we intend to organize the class of these sets, say C, into disjoint equivalence classes using the appropriate equivalence relation, let us decide the following problems.
1) Which equivalence relation ∼ may be defined for C to enable of comparing even these sets, which have (at most) finite common intersection? 2) Assuming that [ ). 3) Assuming that all sets from C organized by ∼ create a quotient algebra being a partial order of exactly 13 elements -establish the maximal possible cardinality of anti-chains if there are four chains in this structure. How does their cardinality increases if this quotient algebra has 100 elements and 11 chains 8 ? 4) Can we automate the verification procedures for belonging to the equivalent class of finite sets or a ball of a given radius? Which conditions must be satisfied for the task feasibility?

III. THE MULTI-FUZZY NATURAL TRANSFORMATIONS WITH FINITE RELATIVE SETS -IN A MORE AND LESS IDEALISTIC SCENARIO
In a case of a non-fuzzy natural transformation, the diagram commutativity condition (see: point b) of Definition 4) warranties that no relative set arises between the composition gives the same result as the 'download' one, i.e.,, A radically different situation holds for fuzzy natural transformation, when different non-empty relative sets may arise as the difference between the appropriate 'upward' and 'downward' composition mappings. The non-empty relative sets may arise as either the difference set A quite similar situation holds for the multi-fuzzy natural transformations. (See: [23].) Unfortunately, the approach to the multi-fuzzy natural transformation from [23] seems to be too idealistic. Indeed, it only refers to the following two situations: the 'upward' mapping composition contains the 'download' mapping composition, or the inverse inclusion holds.
Against these circumstances, this chapter will elaborate on a new, less idealistic approach to the multi-fuzzy natural transformations. We intend to do it by relaxing the requirement of strict inclusions between (set-theoretic results of) the 'upward' and the 'downward' mapping compositions. It means that we admit the situation when the results differ arbitrarily. However, the finite relative sets will constitute our particular focus area.

A. THE MULTI-FUZZY NATURAL TRANSFORMATIONS WITH FINITE RELATIVE SETS -THE MORE IDEALISTIC APPROACH
The typology of the multi-fuzzy natural transformations -as proposed in [22], [23] -contains two main types of them: the 'upward' and the 'downward' multi-fuzzy natural transformations.' Informally speaking, we deal with the upward ones when the 'upward' diagram composition of mappings constitutes a 'greater' set than the set obtained via the 'downward' diagram composition. If the 'downward' composition generates a set with greater cardinality than the 'upward' composition -we deal with the 'downward' multi-fuzzy natural transformations. If the relative set (as a Formally, they are introduced as follows. Definition 11 (The Upward Multi-Fuzzy Natural Transformation 9 ): Let us assume that a finite sequence of categories C i with two corresponding sequences of functors operating between the categories are given, for a finite k and i = 1, 2, . . . , k. Then the upward forms a family of mapping such that the following requirements are satisfied: • it holds the upward fuzzy commutativity: (the result set of the 'upward' functor composition contains the result set of the 'downward' composition of the functors.) If a relative set, say A, determined by inclusion (3) has a finite cardinality (i.e.,, card(A) < ∞), then we deal with the finite-valued upward multi-fuzzy natural transformation. The same ideas constitute a construction basis not only for further variants of the upward multi-fuzzy natural transformation but also for a variety of their downward counterparts; thus, we can omit details of their constructions. 10 It is difficult to resist the impression that the construction of the taxonomy of the multi-fuzzy natural transformations relies on some idealization. Indeed, it was idealistically assumed that a result of the 'upward' composition of mappings is included in a result of the 'downward' composition, or it holds the inverse inclusion. It refers to relatively rare relationships between sets. It motivates us to consider a broader spectrum of possible situations.

B. THE MULTI-FUZZY NATURAL TRANSFORMATIONS WITH FINITE RELATIVE SETS-TOWARDS A LESS IDEALISTIC APPROACH
Meanwhile, one can imagine a more general situation, when the sets as results of the 'upward' and 'downward' mapping compositions are almost identical, but neither the first set is included in the second one, nor the second one in the first one. The condition that their symmetry difference (symbolically ) is finite may formally depict the situation. More formally, we are interested in the case when the symmetric difference of the compositions of functors operating between the categories are given, for a finite k and i = 1, 2, . . . , k. Then the 10 They may be easily found in [23].

multi-fuzzy natural transformation with the (almost) commutativity condition
forms a family of mapping such that the following requirements are satisfied: • it holds the fuzzy (almost) commutativity: where 'fin' denotes a class of finite sets (of a given space, universe, domain, etc.). We will denote this type of relative sets as in (5) in terms of = -read 'almost equal' (Fig. 4): Let us assume that the condition of finite symmetry difference (the = -type relation between different composition sets) enables of introducing some unification and avoiding the dichotomy between the 'upward' and 'downward' variants of the multi-fuzzy natural transformation from [22], [23]. Simultaneously, it also delivers a way to modify the concept of a multi-similarity from [23] These three properties of the unification-based approach to the multi-fuzzy natural transformation from Definition 15 will be developed in chapter IV. In its framework, we intend to introduce a piece of algebraic machinery to organize the entire spectrum of the relative sets created at each stage of the diagram construction for the multi-fuzzy natural transformations. The pairs of them, which has finite symmetry difference, will constitute a particular focus area of our interests.

IV. THE MULTI-FUZZY NATURAL TRANSFORMATION WITH RELATIVE SETS IN P(ω)/fin
Let us repeat that the entire spectrum of the relative sets is the effect of the diagram construction for the multi-fuzzy natural transformation. We intend to organize a spectrum of these sets. The algebraic structure -exploited in it -will be a unique P(ω)/fin, the so-called composition algebra P Comp (ω)/fin.
In order to introduce this structure in the categorial contexts of the multi-fuzzy natural transformations, let us assume that the following 'upward' and 'downward' mapping composi- Let us now introduce the following ∼ fin relation between these mapping compositions.
We will prove that ∼ fin constitutes an equivalence relation. For simplicity of further explanation, let us divide the class of all mapping compositions into the 'upwards' and the 'downward' group and denote them by Comp U and Comp D (resp.).
The elements of Comp U and Comp D , and the methods of their creation are collected in the following table created for the situation presented in Fig. 4. (For editorial reasons, we will use the abbreviation H i for Hom i in the table, i = 1, 2, . . . , k − 1.) 11 It is noteworthy to make a methodological remark in this place. Although we define the classes Comp U and Comp D as the collections of composition mappings, we will also have a tendency to treat them in a more set-theoretic way, i.e.,, as a collection of the composition mapping values. The interpretation method will stem from the context of analysis. 11 Although both Comp U , Comp D are considered as finite classes, at each k, it is admissible to consider them as potentially infinite. It is not difficult to show that ∼ fin ∈ Comp U × Comp D (defined as above by the formula taken from condition (5)) forms an equivalent relation and divide the class Comp U × Comp D of all composition mappings over a given multi-fuzzy diagram into disjoint abstract classes. We propose to prove this fact in the form of the following lemma.
fin ⇐⇒ y x ∈ fin ⇐⇒ y ∼ fin x), thus ∼ fin is symmetric. 12 3) Transitivity follows from the fact that for each sets A, B, C: Since A B ∈ fin and B C ∈ fin, it implies that also A C ∈ fin. In particular, ∀x, y, z ∈ Comp * (x ∼ fin y ∧ y ∼ fin z ⇒ x ∼ fin z). Instead of Comp * , we will sometimes write P Comp (ω) to underline the set-theoretic aspects of mapping compositions, more as mappings themselves, but as the sets of their values. The set-theoretic perspective in defining of P Comp (ω) 12 One can also infer this property immediately from commutativity of symmetric difference. VOLUME 10, 2022 finds its reflection in the concept of the composition quotient algebra P Comp (ω)/fin, which stems from the concept of P Comp (ω).

A. P Comp (ω)/fin FOR RELATIVE SETS AND ITS ALGEBRAIC FOUNDATION
The nature of ∼ fin as an equivalent relation allows us to introduce a new quotient algebra to be called a composition quotient algebra P Comp (ω)/fin for relative sets. This quotient algebra constitutes a unique subalgebra of the so-called algebra P(ω)/fin. 13 Formally, it is defined as follows.
Definition 14 (A Family of Relative Sets): Let us assume that Comp * is given as associated to a multi-fuzzy natural transformation diagram. Each set A ∈ fin such that for some B, C ∈ Comp * is said to be a relative set for the pair (B, C). A class of all such relative sets is said to be a family of relative sets and denoted by Comp * R . Example 5: Let us establish that for some functors F 1 , F 2 , H 1 (c, −), and the natural transformation components η 0 , η 1 , and let us assume that Then the quotient algebra is said to be a composition algebra of relative sets, and denoted by P Comp (ω)/fin.
Obviously, for each A ⊆ Comp * , we define the equivalent class of this algebra as follows.
In other words, • B belongs to the equivalent class determined by A and ∼ fin if A = B, or the symmetric difference A B is finite, and 13 We omit a detailed explanation of the fact. The concept of the quotient algebra P(ω)/fin itself and its properties -the more and less advanced may be found, for example, in [29]. A nice and compact introduction to this concept may be found in the initial pages of [30]. 14 One can show that it forms an ideal in the class of all subsets of ω.
• P Comp (ω)/fin is created by all equivalent classes determined by sets from Comp * (as the class representatives) and the equivalent relation ∼ fin between them. In order to introduce a piece of dynamism to P Comp (ω)/fin, and to grasp its algebraic nature, we associate two following operations + and · into P Comp (ω)/fin.
To perform the task -let us assume that Comp * constitutes a ρ-algebra, i.e.,, it forms a collection of subsets of ω, which is closed under sums ∪ and complements /. 15 It allows us to define the required operations in the quotient P Comp (ω)/fin as follows. and for A, B ∈ Comp * . Example 6: 1) Due to (7) and Definition  : B Comp * ∈ fin}. Obviously, the only subclass of Comp * , which satisfies the condition, is the class of its infinite (denumerable) subsets, i.e., [Comp * ] = ω/fin. In order to extract a couple of more advanced algebraic features of P Comp (ω)/fin let us enlarge the list of possible -relations on relative sets A, B ∈ Comp * by introducing: These -relations between relative sets find their reflection in the corresponding relations =, ≤, <, · between abstract classes for the sets A, B from Comp * . More precisely, for all A, B ∈ Comp the following connections between these two classes of relations hold: 15 It means that 1) if A, B ∈ , then A ∪ B ∈ , and 2) if A ∈ , then also ω/A ∈ .
where 0 is a zero of P Comp /fin, 16 and the order ≤ in P Comp (ω)/fin is determined as follows Since (A ∪ B) B = A/B, we can alternatively write It is noteworthy to observe that the relation ≤ between the equivalent classes from P Comp (ω)/fin -expressible in terms of ⊆ -enables of defining our quotient algebra as an ordered structure (P Comp (ω)/fin, ≤). It has some interesting algebraic properties. One of them is communicated by the following fact.
Proposition 1: P Comp (ω)/fin has atoms. 17 Proof: Obviously, (P(ω) Comp /fin, ≤ ) as a partially ordered structure is finite. In fact, card(Comp * ) < ∞ and only infinite subsets from Comp * are capable of creating new equivalent classes -different than the class fin. Obviously, the number of these is no greater than card(Comp * ), thus card(P(ω) Comp /fin) < ∞, too. The thesis follows now from the fact that each finite and partially ordered structure is atomness.
Obviously, P(ω) Comp /fin just defined is embedeable in P(ω)/fin because we consider Comp * as finite (even if arbitrarily large). It is noteworthy to state that (even if) we admit Comp * to be infinite (denumerable), P(ω) Comp /fin will constitute some substructure of P(ω)/fin because only a selection of subsets of P(ω) will be contained in it.
Although the assumption on infiniteness of Comp * seems to be a convenient bridgehead to incorporate some results of cardinal arithmetic for P(ω)/fin, such as a serious of theorems concerning the existence of the so-called Hausdorff gaps and limits in it, we will omit their considering for a cost of exposing some combinatorial features of P Comp (ω)/fin and for a cost of extending the previous analysis thanks to a further specification of the concept of the multi-similarity. This issue is the subject of the following subsection. 16 As usual, we adopt the standard convention to define 0 of P Comp /fin as [∅] ∼ fin . This solution allows us to establish a homomorphism: P Comp (ω) → P(ω)/fin. 17 An atom in a given ordered structure (A, ≤) is a minimal non-zero element of this structure.

B. P Comp (ω)/fin AND THE RECAPITULATED IDEA OF k-MULTI-SIMILARITY BETWEEN SETS
The analysis of the previous subsection delivers only a general method (or a frame) to organize the internal structure of P Comp (ω)/fin and the mutual relations between relative sets -obtained at each construction stage of a given multi-fuzzy natural transformation diagram.
This general approach is not sensitive to the differences between finite sets (all of them fall into the same equivalent class fin); thus, it is hardy exploitable in computational contexts. Therefore, arises a need to specify the relations slightly more. It will be executed by further specification the concept of multi-similarity towards the concept of a k-multi-similarity for a fixed natural k. It delivers a piece of more substantial knowledge about similarities between two given sets.
In this chapter, k-multi-similarity is firstly introduced for sets from P Comp (ω). Secondly, we propose to extrapolate it for equivalence classes from P Comp (ω)/fin. It allows us to define the so-called k-similarity balls as a notion conceptually close-related to the concept of Hamming's balls. Being equipped with these two concepts, we can computationally explore the internal structure of P Comp (ω). It will be a matter of the first subsection of this section. Similarly -the analogy notions of abstract k-similarity and the abstract k-similar balls allow us to computationally explore the internal structure of P Comp (ω)/fin. This task will be the subject of 2. subsection of this section.

Definition 16 (The k-Multi-Similarity up to the Difference Set A): Let K , M be two arbitrary sets. It will be said that K is k-multi-similar to M (or M is k-multi-similar to M ) up to d.s. A if and only if
• card(K M = A) = k, and k is finite. We will write K ∼ M (M ∼ k K resp.) up to d.s. A.
This idea may be incorporated for defining the so-called finite sum k-multi-similarities and for the finite intersection k-multi-similarities. These definitions may be completed by the concept of k-similarity ball.

Definition 19 (k-Similarity Ball): Let A ⊂ P Comp (ω) be an arbitrary set, and k be a non-negative integer. The set
B k (A) := {B ∈ P Comp (ω) : |A B| ≤ k} is said to be a k-similarity ball of a center A and radius k.
It allows us to formulate the following theorems. It states that no k −1 similar ball is k-multi-similar. Informally speaking, we can always find such a finite sum (of some subsets from the P comp (ω) universe), which are 'too far from the ball center A. 18 In other words, they differ from the ball center A more than it is admissible by the ball radius.
Theorem 1: Proof: The goal of the proof is to show that we can always find such a finite sum of subsets of P comp (ω), which is further from a fixed ball center A than admissible value k − 1.
For that reason, let Y ⊂ P Comp (ω), and |Y | = k. Let S = ∪ n i=1 S i , i.e., let S be an arbitrary finite sum of some To prove the thesis, it is enough to compute the symmetric difference ('distance') ∪ n i=1 S i A now. From 1 and 2 -we obtain the inequality: Proof: The proof runs as previously. It is only enough to consider the finite intersections instead of finite sums in the whole reasoning.
One needs to underline that we implicitly assume that the finite sums and intersections create non-empty sets. Otherwise, the situation from the theses of theorems cannot hold (empty set belongs to each ball).

2) THE IDEA OF AN ABSTRACT k-MULTI-SIMILARITY FOR EQUIVALENCE CLASSES OF P(ω) Comp
The k-multi-similarity is suitable to measure distances between sets from P Comp . The intention of the chapter is to extrapolate this measurement idea for equivalence classes as elements of P Comp (ω)/fin. For that reason, the definition of 18 |A| denotes a cardinality of A.
the abstract k-multi-similarity and the concept of the abstract k-similarity ball will be introduced. Nevertheless, one may have a feeling that P Comp (ω)/fin requires further specification to be a suitable structure for reconstructing a piece of classical combinatorics in its context. In particular, the class 'fin' of P Comp (ω)/fin forms a unitary object from the perspective of the general definition of this algebra (Definition 15), and it should be specified in a more detailed way to determine a formal environment for introducing a piece of combinatorics to P Comp (ω)/fin. From the same 'finite' perspective, the relations (11)-(13) require some unique modification. 19 Thus -instead of = , ⊂ and ⊆ relations -we will consider their modifications: = ≤k , ⊂ ≤k and ⊆ ≤k (resp.) for a fixed natural k.
Definition 20: For all sets A, B ∈ P Comp (ω), and an arbitrary, but a fixed natural k, we define: A ≤k B ⇐⇒ card(A/B) ≤ k and card(B/A) > k, (22) In fact, these -type relations (21)-(23) play a slightly technical role to help us extrapolate the idea of relations for the elements of P Comp (ω)/fin, i.e., for equivalent classes. It is not difficult to see that = -relation is weaker than the ordinary = relation. It follows from this simple fact that two sets are recognized as identical in the sense of = -relation even if they are differentiated by a finite set. Similarly, ⊆ , and are weaker than the ordinary ⊆, and relations. These arrangements allow us to specify the internal structure of P Comp (ω)/fin (of the 'fin' class, in particular), and to introduce an order relation between them. In order to perform these tasks -we introduce a new equivalence relation k inside the 'fin' class. More precisely, we will impose on k a requirement of being a congruence. It gives us a warranty that each algebraic operation of equivalence classes (elements of 'fin') will be preserved by this operation. The formal interpretation of this fact is given by definition.
A n → A} 20 be an algebraic system. The equivalence relation ∼⊂ A × A is said to be a congruence if ∀a 1 , . . . a n , b 1 , . . . b n ∈ A the following condition holds: Example 8: Let us establish a natural n > 1, and let us assume that the following algebraic system A = 19 Indeed, they seem to be enough from the perspective of a cardinal arithmetic, but are less suitable from the operational perspective in finite domains. 20 It easy to see that we can consider f A as a multi-argumental automorphism, and -for a generality of the depiction -no further conditions are imposed on it. We omit a detailed definition of the notion of the algebraic structure of a given signature as slightly redundant from this point of view. It may be found in each handbook of general algebra, such as [13] and many others.

{(Z, +, −, ·), } is given, where (Z) is a set of integers, and +, −, · are standard arithmetic operations on pairs of elements from Z. Let us finally establish that the relation n holds between two a, b ∈ (Z) if and only if
It is not difficult to see that not only n is an equivalence relation, but also it forms a congruence on A. Considering (modules of) a, b s as cardinalities of some subsets from Z, one can also associate the n congruence with finite subsets as elements of P(Z).
It allows us to note the following obvious fact. Fact 1: Let us assume that P Comp (ω)/fin -as determined by Definition 15 -is given. Let also 'fin' denote the ideal of all finite subsets of ω, i.e., fin = [∅] ∼ fin . Let also assume that ⊂ A × B, for A, B ∈ fin is a congruence. Then the equivalence class 'fin' forms a quotient structure consisting of a pairwise disjoint equivalence class.
Proof: It follows from the definition of each congruence as a unique equivalence class and from the abstraction principle.
Being equipped with the definitions of ⊆ ≤k -type relations, the definition of congruence and its properties, we are in a position to introduce the corresponding relations = ≤k , ≤ ≤k , < ≤k between the equivalence classes -newly created by congruence on [∅] ∼ fin , e.g. the 'fin' class of P Comp (ω)/fin. In general, the following situations should be distinguished: 1) when the sets A, B ∈ [∅] ∼ fin (i.e., both are finite), and 2) when A, B are such that [A] ∼ fin = [B] ∼ fin . 21 In general, we intend to express the new relations (i.e., = ≤k , ≤ ≤k , < ≤k ) between equivalence classes determined by a congruence, say , in terms of the old ⊆ ≤k -type relations.
Since the existence of such relations between two equivalence classes should be independent of a choice of the relation representatives, it forces a need to use general quantifiers in explanans of the appropriate definition of these relations. This postulate is reflected in the following definition.

⇐⇒ ∀ C∈[A] A ∀ D∈[B] B
Before exemplifying the definition, it is reasonable to make a couple of explanatory remarks.
1) The common sense of clause b) of 2) may be explained as follows. If we take two sets, A and B from two different ∼ fin -equivalence classes, and we intend to check whether they remain in ≤ ≤k -relation, then we should only check whether the corresponding ⊆ ≤ holds between all sets-representatives of the classes  23 Having already defined the new relations -we can venture to extrapolate the idea of k-multi-similarity for sets of P Comp for equivalent classes in P Comp /fin. In this way, we introduce a new type of k-multi-similarity to be called abstract k-multisimilarity, and the abstract k-similar balls. Finally, it allows us to formulate and prove the fact that (k − 1) -balls are not abstract k-multi-similar. Whereas, the k-multi-similarity and the abstract k-similarity ball may be proposed for the equivalence classes of P Comp /fin, i.e., with respect to ∼ fin relation, we will decide to introduce them for the equivalence classes inside the equivalence class 'fin,' i.e., with respect to a congruence inside [∅] ∼ fin . It reflects the idea of the distance measurement for finite sets only.
for a fixed k.
It is easy to see that the abstract k-multi-similarity is based on the mutual relationship between [A] and [B] as in Definition 22 (2b)). It only extracts it in another way.
It is also clear that the situation when [A] = [B] is just expected for the proper definition of the abstract k-multisimilar balls. In fact, it forms a necessary condition to introduce non-trivial balls with a non-zero radius. 24 They are defined as follows.   24 In practice and for some convenience -a natural value as a radius is considered. Thus, we use k instead of r to denote the ball radius. It is noteworthy to observe that this result forms a far and general echo of the well-known result from the error encoding theory, which may be expressed as follows. If a word b does not belong to the (Hamming) ball of a radius r, and a center in a given word a, then the transmission of a with at most d errors does not enable of achieving the word b. Obviously, a difference between these two situations manifests itself in the difference between the entities or objects that participate in the procedure. (See: [21], pp. 376-377.) In the encoding theory-determined situation, we deal with words (as finite sequences). In the conceptual 'entourage of the paper analysis -we deal with equivalence classes determined by sets and the equivalence relation ∼ fin defined in terms of asymmetry difference as previously.
Remark 1: It seems that the thesis of Theorem may be generalized for equivalence classes of the entire P(ω)/fin determined by ∼ equivalence relation, i.e., for all [A] ∈ P(ω)/fin, the ball B k−1 ([A]) is not abstract k-multi-similar.
Reasoning as previously for k-multi-similarity, we could find such an equivalent class [D], for some set D ∈ P Comp (ω)/fin, that A D ≥ k, i.e., D does not belong to the ball In order to grasp different combinatorial properties of P Comp (ω)/fin algebra, we will consider it as the ordered structure (P Comp (ω)/fin, ≤ ). It has been already established that (P Comp (ω)/fin, ≤ ) is atomness. One could extend the proof argumentation line in order to extract the fact that (P Comp (ω)/fin, ≤ ) -as partially ordered structure -cannot constitute any linearly ordered structure. Indeed, if it is possible, all anti-chains 25 in (P Comp (ω)/fin, ≤ ) should form (at most) singletons. (It exactly means that each element of this structure may be compared with any other in the sense of ≤ .) Meanwhile, anti-chains in (P Comp (ω)/fin, ≤ ) may contain more elements -due to the condition defining ≤ -order. In fact, if A, B ∈ {Comp * } are infinite, then each of them creates the same abstract class with all these infinite subsets 26 which create only finite relative sets with it, i.e., it holds A/C j ∈ fin, B/D k fin and for each j ∈ J , k ∈ K . Simultaneously, if A, B do not satisfy the same condition, i.e., A/B ∈ fin, then also [A] ≤ [B]. Thus, the algebra cannot be linearly ordered structure, but it forms a proper partially ordered structure.
This fact determines a non-triviality of different combinatorial features of (P Comp (ω)/fin, ≤ ). In order to illustrate this conjecture, we propose to prove the theorem about cardinalities of chains and anti-chains of nm + 1-elemental (P Comp (ω)/fin, ≤ ). It requires a unique version of dual Dilworth's Theorem. 27 Theorem 4 (A Dual of Dilworth's Theorem for (P Comp (ω) /fin, ≤ )): The size of the largest chain in (P Comp (ω)/fin, ≤ ) equals the smallest number of anti-chains into which the (P Comp (ω)/fin, ≤ ) may be partitioned.
Theorem 5: Let us assume that (P Comp (ω)/fin, ≤ ) has exactly nm + 1 elements, for some fixed n, m ∈ N. Then (P Comp (ω)/fin, ≤ ) contains a chain of the length n + 1 or an anti-chain of the length m + 1.
Proof: Let us assume that no chain of the length n + 1 exists in (P Comp (ω)/fin, ≤ ). We will show that there exists an anti-chain of the length m + 1. Meanwhile, the dual Dilworth's theorem allows us to write (P Comp (ω)/fin, ≤ ) = ∪ n i=1 A i , i.e., represent the algebra as a sum of n antichains. Since nm so it must be |A m | ≥ m + 1, for some natural m. Otherwise, if ∀i ∈ 1, . . . , n, |A i | is at most m, then the inequality (25) cannot be satisfied. The proof of dual Dilworth's Theorem delivers a general framework of the operational method of achieving the anti-chain partition of P Comp (ω)/fin. Indeed, the main idea is to take the minimal elements of the newly created subsets of P Comp (ω)/fin. The method of their construction is simple: at first we take the minimal elements of P = dom(P Comp (ω)/fin), and take them as the first anti-chain, say A 1 . Secondly, we take the minimal elements of the difference set P−A 1 as the second anti-chain, say A 2 . Next, the third antichain, A 3 , is(are) the minimal element(-s) of P−(A 1 ∪A 2 ), etc. In this way, we can achieve an anti-chain partition of P, i.e., P = A 1 ∪ . . . ∪ A h . The exact order of the steps is presented by Algorithm 1.

V. SOLVING THE LEADING PROBLEMS
In Section I, the following questions have been formulated.
1) Which equivalence relation ∼ may be defined for C to enable of comparing even these sets, which have (at most) finite common intersection? 26 J and K form a finite set of indices. There is no need to specify it more. 27 We omit its proof as it runs in a standard version of the theorem. The proof of Dilworth's Theorem and its dual may be found in each handbook of combinatorics. See, for example: [21]. while j > 1 and set ⊂ P do 6: if j := 2 then 7: set := P−A 1 ; 8: end if 10: if j := 3 then 11: set := P − (A 1 ∪ A 2 ); 12: 13: end if 14: if j := 4 then 15: set := P − (A 1 ∪ A 2 ∪ A 3 ); 16:

18:
if j := h then 19: set end if 22: end while 23: Being equipped with the results, just elaborated, we can venture to formulate the answers to these questions.
Ad. 1: Due to arrangements from Section IV -the class C of all pairs (A, B) constitute a unique quotient algebra P comp (ω)/fin determined by the following equivalence binary relation ∼ fin between two composition sets (results of 28 Obviously, the numbers of chains and anti-chains play an exemplary role only. VOLUME 10, 2022 the 'upward' and 'downward' composition mappings): It was also shown that P comp (ω)/fin may be partially ordered by relation ⊆ defined by the condition: which introduces an order ≤ among entities of P comp (ω)/fin due to the relation: Ad. 2 Ad. 3: In Section IV (part C) -by means of dual Dilworth-Theorem -it was also proven that if (P Comp (ω)/fin, ≤ ) has exactly nm + 1 elements, for some fixed n, m ∈ N, then (P Comp (ω)/fin, ≤ ) contains a chain of the length n+1 or an anti-chain of the length m + 1. Hence, if the quotient algebra has 13 elements and 4 chains, then the maximal number of anti-chains might be equal to (13 -1)/4 = 3. Similarly, if the quotient algebra has 100 elements and 11 chains, then the maximal number of anti-chains might be equal to (100 -1)/11 Ad. 4: Whereas the most general and theoretic constructions of the approach (such as constructions of P Comp (ω)/fin, its equivalent classes, the class of finite sets, say FIN, as a quotient set, etc.) often resist the effort of their programmingbased depiction, one can venture to automate some verification procedures. We demonstrate two exemplary verification procedures in terms of R-language: 1) for belonging to a FIN equivalent class, and 2) for belonging to a given ball of a radius p. In both cases, the appropriate control functions may be defined: 'FIN-cont' and 'BALL-cont3' as depicted in Fig. 5 and Fig. 6 (resp.) for the task performing.
In the first case, we consider the FIN-class as restricted to a given natural p (i.e., as a class of all finite sets with cardinalities up to p) 29 The FIN-cont function is defined by the if-else instruction: if cardinality of a symmetry difference of two sets -denoted by variables x and y -is no greater than p (the blue marked line), then it is confirmed that the sets belong to FINp. Otherwise -they do not. The if-else instruction also defines the 'BALL-cont3' function. The core line (blue marked) of the definition body for the function introduces a sum of cardinalities 30 29 The main idea is to consider p as an arbitrary natural number. 30 In practice, we identify cardinalities of finite sets with lengths of the vectors representing the sets.   i, j, k = 1, 2, 3). If the entire sum is no greater than 3*3*p, then classes x, y, z belong to pBall. 31 Otherwise-they do not. Fig. 8 illustrates a simplified version of the situation from Fig. 6. -restricted to two equivalence classes (two variables x and y). Fig. 7 delivers an exemplary R-code for FIN-cont and the situation of two 1-set equivalence classes A, B for verification procedure whether these sets belong to FIN3-class. Fig.9 illustrates an exact implementation of R-commend as in Fig. 6 for some sets A, B, C previously defined. 31 This multiplication follows from the fact that we should compute all the 'cardinality distances' between all sets in each equivalence class to multiply them by the accepted distance p for each pair of such sets. Assuming that we have three equivalence classes x, y, z and three elements in each class, we get 3 * 3 * p as the supremum of the accepted values of these cardinal distances.

VI. STATE OF THE ART
As already mentioned, category theory aspires to be a new, functional paradigm in the foundations of the entire area of formal sciences. However, it stems from homology algebra, and its general frame was elaborated in McLane's and Eilenberg's works from the 40s. The idea of functors and the natural transformation itself was elaborated in [1], [2], whereas the concept of duality was introduced in [3]. Independently of the relative early provenance of category theory, a renaissance of research on practice-motivated aspects of it in the frame of computer science coincides both with the development of such programming-wise tools as Haskell and its enhancements (see: [31]), and different attempts of category theory popularization outside mathematics and theoretical computer science as in [4], [5], [27].
By contrast -the proper birth of the algebraic theory of P Comp (ω)/fin is -due to the author's best knowledgedifficult to indicate. Indeed, this idea stems from the classical Haussdorf's papers [18], [19] on the so-called Haussdorf's gaps and limits, and from research on Stone's representation as in [7]. In later decades, a development of the theory of P Comp (ω)/fin algebras was -on the one hand -supported by research on the so-called Parovicenko's algebras, as in [20]. On the other hand, a new catalyzation of this development was the idea of forcing -broadly exploited in research on the algebras in such sophisticated works as [12], [17] and in many others. Although some metamathematical monographs on Boolean algebras, such as: [14], [15] treated the issue perfunctorily (if any), P Comp (ω)/fin returned to scientific court in [13] and in many other works, such as [17], [32]. It would be challenging to discuss the entire spectrum of directions and contexts in which these algebras are immersed and described in scientific literature. It is not coherent with the main task of the paper. One only needs to underline that all the directions have been developed independently of the category theory in a purely set-theoretic paradigm. Indeed, the theory of P Comp (ω)/fin algebras -as an integral part of the theory of gaps and limits -is immersed in the arithmetic of high cardinals. As such one -it has almost nothing to do with classical finite combinatorics -as described in [21].
Against these tendencies -another approach to P Comp (ω)/fin was elaborated in this paper. Its first goal is not to elaborate some new metamathematical properties of these algebras in terms of the arithmetic of high cardinalities but to organize the spectrum of relative sets in the context of the multi-fuzzy natural transformations. As previously indicated, this approach was incorporated to build some new connections with classical encoding theory. Finally, this paper research breaks the common tendency to exploit the Haskell language in a role of programming-wise support for category theory -although Haskell's functionalities and the entire philosophy of programming in Haskell is strongly motivated by an algebraic tissue of category theory. The reason for the solution has already been explained in 'Introduction.' This paper may be viewed as a development of considerations from [22], [23], [28]. However, the condition of the diagram commutativity is violated in the paper approach in another way. Here, we consider pairs of almost equal relative sets, while the appropriate inclusions between the sets from the pairs were required in the previous papers.

VII. CONCLUSION AND CLOSING REMARKS
By introducing a piece of machinery of the Boolean algebra theory -we showed how to organize the spectrum of the so-called relative sets due to fuzzifying the commutativity condition for the natural transformation diagrams. We have initially introduced an equivalence relation ('identifying' two sets if they have a finite common intersection). Secondly, we collected the equivalent classes into a quotient structure called the composition algebra for relative sets. Next, we equipped this algebra with the appropriate order relation to organize it and extract a piece of algebraic and combinatorial properties of this structure. In this way, we have also illustrated how one could reconcile the dynamic, categorial paradigm with the purely set-theoretic one -the seemingly unreconciliable paradigms. It seems that this project could be performed thanks to our decision to treat the class Comp * more set-theoretically -as values of the mapping compositions than the mappings compositions themselves. It allowed us to break the difficulty with the different nature of these two approaches.
From a broader perspective, we can specify the paper's analysis as a 'reflection on differences' (between the 'upward' and the 'downward' mappings compositions), which introduces a piece of fuzziness to the multi-natural transformations. It is noteworthy to underline that the formal algebraic apparatus enabled considering pairs of the sets with only a finite common intersection. Simultaneously, one needs to underline that only several conceptual 'bridgeheads' have been constructed in the area of a typical interaction of these two paradigms for the use of the analysis of the paper. Fortunately, their construction delivers a piece of optimism regarding the possibility to reconstruct a more significant portion of encoding theory in the conceptual framework of the analysis. In particular, it seems that the mutual relationships between the idea of distance measurements in terms of k-multi-similarity and k-multi-balls and the same idea of distance measurement in terms of Hamming's distances may be deeper emphasized in the future. It seems to be a promising subject of further analysis.