On the Digital Pontryagin Algebras

In the current study, we explore digital homology modules, and investigate their fundamental properties on (pointed) digital images as one of the developments of symmetries. We also examine pointed digital Hopf spaces and base point preserving digital Hopf functions between pointed digital Hopf spaces with suitable digital multiplications, and explore the digital primitive homology classes, digital Pontryagin algebras on digital Hopf spaces as a symmetric phenomenon in mathematics and computer science.


History and Hopf Space
Digital geometry deals with bounded and finite discrete sets in the sense of classical topology, which is considered to be digital images or digitalized models of bounded and finite subsets of the lattice points in Euclidean space. The homology modules, higher homotopy groups, stable homotopy groups and equivariant homotopy groups are useful algebraic and topological tools to solve a large number of problems of algebraic geometry and algebraic topology. In the same lode, the digital counterparts of classical homology modules can be important gadgets to classify (pointed) digital images from the point of view for the digital version of the homotopy type, mathematical morphology, and image synthesis. In particular, the informal definitions of many terms in elementary homotopy and simplicial homology theory based on a digital picture on Z 2 or Z 3 were nicely presented in [1][2][3][4][5]; see also [6] for digital quasi co-Hopf spaces.
In the 20th century, a lot of interesting and remarkable results on Lie groups and (pointed) Hopf spaces, as the Eckmann-Hilton dual notions of (pointed) co-Hopf spaces, have been widely investigated and suitable methods have been developed for CW-spaces and usual topological spaces. The (pointed) Hopf spaces were the direct outgrowth of compact Lie groups in classical homotopy theory, as described in [7][8][9]. Indeed, a pointed Hopf space is a triple (Y, y 0 , m Y ) which consists of a pointed topological space (Y, y 0 ) and a base point preserving continuous multiplication m Y : Y × Y → Y such that a constant function e y 0 : Y → Y at y 0 plays a role of a homotopy identity, i.e., m Y (e y 0 , y) = y = m Y (y, e y 0 ) for all y ∈ Y in the pointed homotopy category. The notion of (pointed) Hopf spaces is one of the Eckmann-Hilton dual notions of a co-Hopf space; see [10][11][12][13][14][15][16][17][18][19][20][21][22] for topics related to those basic notions. It can be seen that all (pointed) Lie groups are (pointed) Hopf spaces. In general, the Hopf spaces lack associative and inversive properties and do not have the structure of usual topological manifolds at all.
Multiplication in a Hopf space provides the homology modules of a Hopf space with an algebraic structure which is natural with respect to Hopf functions. In fact, we can construct the algebra structure at homology level which is called as the Pontryagin algebra of the Hopf space. Under suitable conditions, the diagonal function gives the non-negatively graded homology module a coalgebra structure. The two algebraic structures are related to each other and covert the non-negatively graded homology modules into a classical Hopf algebra.

Motivation
There are a few standard approaches for considering a digital analogue of the well-known usual topology on R n such as the graph-theoretic approach, the imbedding approach, and the axiomatic approach [23]. From this point of view, we need to investigate another approach to study digital topology out of classical algebraic topology. In the present paper, we introduce another consideration of a digital analogue as the so-called algebraic approach from the classical homology and Pontryagin algebra. More precisely, the current study is concerned with setting up more algebraic invariants and their fundamental properties of digital homology modules over a commutative ring with identity for digital image with an adjacent relation which are based on the classical homology groups of topological spaces in mathematics and computer science.

Organization of the Paper
The current paper has been organized as follows. In Section 2, we introduce the general notions of digital images with k X -adjacent relations. In Section 3, we consider a digital n-simplex, digital n-chains, and digital homology modules over a commutative ring R with identity of digital images. We also investigate some fundamental and interesting properties of digital homology modules and primitive homology classes of digital images. In Section 4, we consider a pointed digital Hopf space together with digital multiplications, digital homotopy associative and commutative multiplications, and base point preserving digital Hopf functions between pointed digital Hopf spaces with digital multiplications based on pointed digital sets. We also explore important properties of digital primitive homology classes and digital Pontryagin algebras in digital Hopf spaces as a symmetric phenomenon [24] in mathematics and computer science.

Preliminaries
Let Z be the ring of integers and R the field of all real numbers. Let Z n be the set of all lattice points in the n-dimensional Euclidean space R n . A digital image is a pair (X, k X ), where X is a bounded and finite subset of Z n R n and k X indicates some adjacent relation between the members of X; see below.
For an integer u with 1 ≤ u ≤ n, we will first define an adjacent relation of a digital image in Z n as follows. Definition 1 ([25]). Two points p = (p 1 , p 2 , . . . , p n ) and q = (q 1 , q 2 , . . . , q n ) with p = q in Z n are k(u, n)-adjacent if (1) there are at most u distinct indices i with the property |p i − q i | = 1; and (2) if |p j − q j | = 1, then p j = q j for all indices j.
A k(u, n)-adjacency relation on Z n may be denoted by the number of points that are k(u, n)-adjacent to a point p ∈ Z n . Moreover, • the k(1, 1)-adjacent points of Z are called 2-adjacent; and • the k(1, 2)-adjacent points of Z 2 are called 4-adjacent, and the k(2, 2)-adjacent points in Z 2 are called 8-adjacent.
We mostly denote k(u, n)-adjacent relation on a digital image X by k X -adjacent relation for short if there is no chance of ambiguity.
Definition 2 ([26,27]). A digital image (X, k X ) in Z n is said to be k X -connected if for every pair of points {x, y} ⊂ X with x = y, there exists a set P = {x 0 , x 1 , . . . , x s } ⊂ X of s + 1 distinct points such that x = x 0 , x s = y, and x i and x i+1 are k X -adjacent for i = 0, 1, . . . , s − 1.
The following is a minor modification of an earlier definition of a digital continuous function given in [27] (Definition 2.3); see also [28]. Definition 3. Let (X, k X ) and (Y, k Y ) be digital images with k X -adjacent and k Y -adjacent relations, respectively. A function f : then it is not difficult to show that the composite g • f : X → Z of f and g is (k X , k Z )-continuous. Thus, it is possible to construct the category D of digital images and digital continuous functions; that is, the object classes of D are digital images and the morphism classes are digital continuous functions.
Definition 4 ([25,27,29]). Let (X, k X ) and (Y, k Y ) be digital images with k X -adjacent and k Y -adjacent relations, respectively, and let f , g : X → Y be (k X , k Y )-continuous functions. Suppose that there is a positive integer m and a (k Then, F is called a digital (k X , k Y )-homotopy between f and g, written as F : f (k X ,k Y ) g, and f and g are called digitally (k X , k Y )-homotopic in Y.
We note that an adjacent relation k X×[0,m] Z on the cartesian products X × [0, m] Z in Definition 4 has been used as the so-called generalized normal product adjacency relation. We now describe the pointed versions of digital images to develop the pointed digital category as follows.

Definition 5 ([30,31]).
A pointed digital image with k X -adjacent relation is a triplet (X, x 0 , k X ), where X is a digital image and x 0 ∈ X. In this case, x 0 is said to be a base point of (X, between pointed digital continuous functions f and g is said to be pointed digital (k X , k Y )-homotopy between f and g if F(x 0 , t) = y 0 for all t ∈ [0, m] Z .
We now construct the so-called pointed digital category D * of pointed digital images and base point preserving digital continuous functions; that is, the object classes of D * are pointed digital images and the morphism classes are base point-preserving digital continuous functions.

Digital Homology Modules
In this section, we consider the digital homology modules [32,33] over a commutative ring R with identity '1 R ' (compare with [4] in the case of digital simplicial homology groups).
Unlike the classical convex combination of points, it can be easily verified that x is a digital convex combination of e 0 , e 1 , . . . , e n if and only if x is an element of {e 0 , e 1 , . . . , e n }. We let ∆ n be the set of all digital convex combinations of points e 0 , e 1 , . . . , e n in Z n+1 ; that is, ∆ n = {e 0 , e 1 , . . . , e n }, which is completely different from the usual convex combinations in algebraic topology when n ≥ 1. Considering ∆ n as the digital image with k(2, n + 1)-adjacent relation, we can see that it is k(2, n + 1)-connected, and we call ∆ n a digital standard n-simplex. We denote the k(2, n + 1)-adjacent relation in the digital image ∆ n by k ∆ n for our notational convenience, as mentioned earlier.
where ∆ n is the digital standard n-simplex. Let R be a commutative ring with identity 1 R and let (X, k X ) be a digital image with k X -adjacent relation. For each n ≥ 0, we define dC n (X; R) to be the non-negatively graded free R-module with basis all digital n-simplexes in (X, k X ). The elements of dC n (X; R) are called digital n-chains in (X, k X ).
For each n and i, we now define the i-th face function as the function which would send the ordered vertices {e 0 , . . . , e n−1 } to the ordered vertices {e 0 , . . . ,ê i , . . . , e n } {e 0 , . . . , e i , . . . , e n } preserving the displayed orderings as follows: where the numbers r i are the barycentric coordinates of a point of ∆ n−1 or ∆ n .
Let (X, k X ) be a digital image with k X -adjacent relation, and let σ : (∆ n , k ∆ n ) → (X, k X ) be a digital n-simplex in (X, k X ). Then, the map ∂ n : dC n (X; R) → dC n−1 (X; R) defined as for n = 0 is called the digital boundary operator of the digital image (X, k X ). It can be seen in [32] that The kernel of ∂ n : dC n (X; R) → dC n−1 (X; R) is called the module of digital n-cycles in (X, k X ) and denoted by dZ n (X; R). The image of ∂ n+1 : dC n+1 (X; R) → dC n (X; R) is called the module of digital n-boundaries in (X, k X ) and denoted by dB n (X; R). We note that dB n (X; R) is a submodule of dZ n (X; R) for each n ≥ 0.
The n-th digital homology module dH n (X; R) over R of a digital image (X, k X ) with k X -adjacent relation is defined by dH n (X; R) = dZ n (X; R)/dB n (X; R) for each n ≥ 0, The coset [z n ] = z n + B n (X; R) is called the digital homology class of z n , where z n is a digital n-cycle; see [32] for more details. Let i 1 : X → X × X be the first inclusion, and i 2 : X → X × X be the second inclusion, then we have R-module homomorphisms of digital homology modules induced by i 1 and i 2 , respectively.
We note that dH * plays a role of a bridge between the digital world in computer science and the algebra world in mathematics; see Figure 1.
The digital world in computer science dH * The algebra world in mathematics

Definition 6.
An element x ∈ dH * (X; R) is said to be a digital primitive homology class if where ∆ : X → X × X is the diagonal map.
Let PdH * (X; R) denote the submodule of dH * (X; R) with coefficients in a commutative ring R with identity 1 R consisting of all the digital primitive homology classes. Then, we have the following.
Proof. If x is any digital primitive homology class of dH * (X; R) with coefficients in the commutative ring R with identity 1 R , then from the commutative diagram where i 1 : W → W × W is the first inclusion and i 2 : W → W × W is the second inclusion on W = X or Y; that is, the homomorphic image of the digital primitive homology classes is also digital primitive, as required.

Digital Hopf Spaces and Pontryagin Algebras
From many kinds of algebraic structures in mathematics, we can think of a Hopf group in algebraic topology as a generalization of a usual group in algebra (see [34,35]). For application in computer science, the notions of Hopf spaces or Hopf groups in mathematics will be transformed in this section to those of digital theoretical counterparts in computer science (compare with [36,37]). Definition 7 ([30,31]). Let e y 0 : Y → Y be a constant function at y 0 and let 1 Y : Y → Y be an identity function on Y. A digital Hopf space Y = (Y, y 0 , k Y , m Y ) (sometimes denoted as (Y, y 0 ) for short) consists of a pointed digital image (Y, y 0 ) with an adjacent relation k Y and a (k Y×Y , is commutative up to pointed digital homotopy. Here, k Y×Y is an adjacent relation on Y × Y, and (e y 0 , , and e y 0 is called a digital homotopy identity. As usual, we denote the pointed digital homotopy class by [ f ] as the equivalence class of a pointed digital continuous function f : (X, x 0 , k X ) → (Y, y 0 , k Y ).
where ∆ is a diagonal function.
We now have a symmetric phenomenon in algebra from a digital Hopf space as follows.
If f , g : (X, x 0 ) → (Y, y 0 ) are (k X , k Y )-continuous functions and x ∈ PdH(X; R), then where f * , g * : dH * (X; R) → dH * (Y; R) are homomorphisms of digital homology modules over R induced by (k X , k Y )-continuous functions f and g, respectively.
Proof. We note that and the following diagram is strictly commutative, and similarly for in digital homology R-modules. Since Y has the digital Hopf structure and x is a digital primitive homology class, we have The Figure 2 is a symmetric phenomenon derived from Theorem 1: The following shows that the digital Hopf spaces are closed under the digital products of digital images; that is, the digital Hopf spaces are well behaved with respect to the cartesian products.
Proof. Let m X : X × X → X and m Y : Y × Y → Y be the digital multiplications on X and Y, respectively. We define a function by making the following diagram commute: where S Y×X : Y × X → X × Y is a switching function, and 1 X and 1 Y are the identity functions on X and Y, respectively. Then, for all (x, y) ∈ X × Y, we have Similarly, we also obtain and, in particular, is the switching function. If y is any digital primitive homology class, then, from Theorem 1, we obtain as required.
Definition 10. The digital homology cross product is defined by the homomorphism Under what conditions can we say that the submodule PdH * (X; R) ⊆ dH * (X; R) consisting of digital primitive homology classes is equal to dH * (X; R)? The following gives an answer to this query: Theorem 3. Let dH s (X; R) = 0 for s ≤ n − 1. Then PdH s (X; R) is equal to dH s (X; R) for all s ≤ 2n.
Proof. We consider the Künneth exact sequence of R-modules in algebraic topology; see [34] (p. 228). We note that the Künneth formula is still valid for digital homology modules over a commutative ring with identity 1 R because it is purely algebraic. If m ≤ 2n, then, by our assumption, the torsion part in the above short exact sequence is trivial, so that the cross product is an isomorphism of R-modules. We now consider a homomorphism of R-modules for all m ≥ 0 induced by the diagonal map We see that the target of the R-module homomorphism ∆ * is isomorphic to the first term of the Künneth exact sequence (1). Therefore, for all x m ∈ dH m (X; R), we have for each m ≤ 2n; that is, x m is a digital primitive homology class, as required.
between digital homology modules with coefficients in a commutative ring R with identity 1 R gives an algebraic structure on the digital homology dH * (Y; R).
Then the above algebraic structure on the digital homology dH * (Y; R) is called the digital Pontryagin R-algebra of the digital Hopf space (Y, y 0 , k Y , m Y ).
of digital homology modules over a commutative ring R with identity 1 R . Theorem 4. Let (Y, y 0 , k Y , m Y ) be a digital homotopy associative and commutative Hopf space with a digital multiplication m Y : Y × Y → Y. Then the digital Pontryagin R-algebra dH * (Y; R) becomes a non-negatively graded associative and commutative R-algebra.

Proof. Let
be the digital homology cross product. Then, we can consider the composite of R-module homomorphisms, which we denote by '· P ' or 'm Y * '. Since the digital multiplication m Y : Y × Y → Y is digital homotopy associative, we can see that the following diagram commutes on digital homology R-modules. Moreover, the digital homotopy commutative multiplication on digital homology R-modules, where S * is an R-module homomorphism induced by the switching map S : Y × Y → Y × Y. Therefore, dH * (Y; R) is a graded associative and commutative R-algebra. Example 2. Let X = {(1, 0), (0, 1), (−1, 0), (0, −1)} be the digital image in Z 2 with the 8-adjacency relation as in Example 1. Then (X, x 0 , k X , m X ) is a digital homotopy associative and commutative Hopf space having a digital homotopy inverse, where x 0 = (1, 0) and k X = 8 (see [31] [Example 3.8]). Since (X, x 0 , k X , m X ) has a digital multiplication m X : X × X → X, the algebra structure on dH * (X; R) makes it into the digital Pontryagin R-algebra. Moreover, Theorem 4 asserts that the digital Pontryagin R-algebra dH * (X; R) is a non-negatively graded associative and commutative R-algebra. In particular, if R is a field, then dH * (X; R) is a non-negatively graded vector space of the field R in which we can sometimes use the dimensional issues.
Proof. We need to show that Since it is not difficult to show that the first and second conditions are satisfied, we will only check the final condition. From the following commutative diagram of R-modules, we have as required. induced by f is an R-module automorphism of digital Pontryagin R-algebras.

Remark 1.
Let HD * be the category of pointed digital homotopy associative and commutative Hopf spaces and digital Hopf functions and let PA be the category of non-negatively graded associative and commutative R-algebras and R-algebra maps. Then, Theorems 4 and 5 assert that dH * (−; R) : HD * → PA is a covariant functor.

Conclusions
Hopf spaces and co-Hopf spaces play a pivotal role in algebraic topology, especially in (equivariant) homotopy theory in mathematics. Digital topology deals with the so-called discrete sets with a topology in the sense of Euclidean topology, which is considered to be digital sets or images of finite and bounded subsets of the n-dimensional Euclidean space. In digital topology, the digital process substitutes a bounded and finite discrete set for a suitable object in some category.
In the current study, we have developed the concept of a digital counterpart of classical notions in mathematics using the so-called algebraic approach from the classical homology and Pontryagin algebra. More specifically, this study focused on setting up more algebraic invariants and their fundamental properties of digital homology modules over a commutative ring with identity for a digital image with an adjacent relation which are based on the classical homology groups of topological spaces as a symmetric phenomenon in mathematics and computer science.