In this paper, we recall some definitions and properties from digital topology, and we consider the simplices and simplicial complexes in digital images due to adjacency relations. Then we define the simplicial set and conclude that the simplicial identities are satisfied in digital images. Finally, we construct the group structure in digital images and define the simplicial groups in digital images. Consequently, we calculate the digital homology group of two-dimensional digital simplicial group.
MSC:55N35, 68R10, 68U10, 18G30.
1 Introduction and basic concepts
Digital topology is a branch of mathematics where the image processing and digital image processing are studied. Many mathematicians such as Rosenfeld , Han , Kong [3, 4], Malgouyres , Boxer [6–8], Karaca [9–11] and others have contributed to this area with their research. The notions of a digital image, a digital continuous map and digital homotopy were studied in literature [1, 4, 5, 7, 8]. Their recognition and efficient computation became a useful material for our study.
On the other hand, simplicial groups were first studied by Kan  in the 1950s. We carry the notion of a digital image on to the simplicial groups and construct algebraic structures in digital images such as simplicial sets, simplicial groups by using the figures and definitions in . In dimension one, we use the 2-adjacency relation, and in dimension two, we use the 8-adjacency relation.
Assume that is n-dimensional Euclidean spaces. A finite subset of with an adjacency relation is called to be a digital image.
Let X and Y be digital images such that , . Assume that is a function. Let be an adjacency relation defined on , . Then f is called to be -continuous if the image under f of every -connected subset of X is -connected.
A function which is defined in Definition 1.3 is referred to be digitally continuous. A consequence of this definition is given below.
Let be a finite collection of digital m-simplices, , for some nonnegative integer d. If the following statements hold, then is called a finite digital simplicial complex:
If P belongs to X, then every face of P also belongs to X.
If , then is either empty or a common face of P and Q.
The dimension of a digital simplicial complex X is the largest integer m such that X has an m-simplex.
Example 1.8 If X is a digital simplicial complex and is a -simplex of X, then any subset of is a face of that digital simplex and thus is itself a digital simplex of X. In particular, we can think of the -simplex as a geometric digital simplicial complex consisting of itself and its face.
2 Digital simplicial maps
If K and L are geometric digital simplicial complexes, then a digital simplicial map is determined by taking the vertices of K to vertices of L such that if is a -simplex of K, then are all vertices (not necessarily unique) of some -simplex in L. Given such a function , the rest of is determined by linear interpolation on each -simplex (if can be represented by in barycentric coordinates of the -simplex spanned by the , then ) (see Figures 4, 5).
Corollary 2.1Every digital simplicial map is digitally continuous.
Proof Let K and L be digital simplicial complexes having and adjacency, respectively, and let be a digital simplicial map. Since K is a simplicial complex, then the point of simplices of K have adjacency. So, f maps the points of the simplices of K to the points of the simplices of L. Therefore the image of -adjacent points under f is -adjacent points. Consequently, f is -continuous, i.e., digitally continuous. □
Example 2.2 A simple but interesting and important example is the inclusion of a -simplex into a digital simplicial complex. If X is a digital simplicial complex and is a collection of vertices of X that spans a -simplex of X, then is itself a digital simplicial complex. We then have a simplicial map that takes each to the corresponding vertex in X and hence takes K identically to itself inside X.
Example 2.3 Let be a -simplex, one of whose -faces is . Consider the simplicial map , determined by , , , that collapses the -simplex down to the -simplex (see Figure 5). The great benefit of the theory of digital simplicial sets is a way to generalize these kinds of maps in order to preserve information so that we can still see the image of the -simplex hiding in the -simplex as a degenerate -simplex.
2.1 Face maps in digital images
Given a -simplex, we would like a handy way of referring to its -dimensional faces. This is handled by the face maps. On the standard -simplex, we have face maps , defined so that , where, as usual, the ˆ denotes a term that is being omitted. Thus applying to yields the -face missing the vertex j. It is important to note that each simply assigns one of its faces to the -simplex (see Figures 6, 7).
2.2 Delta sets and delta maps in digital images
Delta sets (sometimes called -sets) constitute an intermediary between digital simplicial complexes and digital simplicial sets.
Definition 2.4 A delta set consists of a sequence of sets which have adjacency respectively, and the maps for each i and , , such that whenever .
Of course this is just an abstraction, and generalization, of the definition of an ordered digital simplicial complex, in which are the sets of -simplices and the are the digital face maps.
2.3 Degenerate maps in digital images
Degeneracy maps in digital images are, in some sense, the conceptual converse of face maps. Recall that the face map takes a -simplex and gives us back its j th -face. On the other hand, the j th digital degeneracy map takes a -simplex and gives us back the j th degenerate -simplex in digital images living inside it.
As usual, we illustrate with the standard -simplex, which will be a model for what happens in all digital simplicial sets. Given the standard -simplex as , there are -digital degeneracy maps , defined by . In other words, gives us the unique degenerate -simplex in with only the j th vertex repeated.
Again, the geometric concept is that can be thought of as the process of collapsing down into by the digital simplicial map defined by for , and for .
This idea extends naturally to digital simplicial complexes, to digital delta sets, and also to simplices that are already degenerate. If we have a (possibly degenerate) -simplex with for each k, , then we set , i.e., always repeats. Thus we may say that this is a digital degenerate simplex in .
Also, as for the , there are certain natural relations that the degeneracy maps possess. In particular, if , then . Note that we have in the last formula, not , since the application of pushes j one slot to the right.
Furthermore, there are relations amongst the face and degeneracy operators. These are a little bit advantageous to write down since there are three possibilities:
These situations are all clear. For example, applying to either side of the first formula yields . Note also that the middle formula takes care of both and .
3 The structure of a simplicial set in digital images
Now we give the definition of digital simplicial sets.
Definition 3.1 A digital simplicial set consists of a sequence of sets which have adjacency respectively, the functions and for each i and with are such that
Example 3.2 If the standard -simplex is now realized as a digital simplicial set, then it is the unique digital simplicial set with one element in each , . Thus the element in dimension n is .
Example 3.3 As a digital simplicial set, the standard ordered -simplex already has elements in each . Namely, .
3.1 Nondegenerate simplices in digital images
A -simplex is called nondegenerate if x cannot be written as for any and any i.
Every -simplex of a digital simplicial complex or digital delta set is a nondegenerate simplex of the corresponding digital simplicial set. If Y is a topological space, then a -simplex of is nondegenerate. It cannot be written as the composition , where π is a simplicial collapse with and σ is a singular -simplex (see Figures 8-10).
Note that it is possible for a nondegenerate -simplex to have a degenerate -face. It is also possible for a degenerate simplex to have a nondegenerate face (for example, we know for any x, it is degenerate or not).
3.2 Categorical definition
As for digital delta sets, the basic properties of simplicial sets derive from those of the standard ordered -simplex. In fact, that is where the prototypes of both the face and degeneracy maps live and where we first developed the axioms relating them. Thus it is not surprising (at this point) that there is a categorical definition of digital simplicial sets, analogous to the one for delta sets, in which each digital simplicial set is the functorial image of a category and Δ is built from the standard digital simplices.
Definition 3.4 The category has the finite ordered sets as objects. The morphisms of are order-preserving functions .
Definition 3.5 (Categorical definition of digital simplicial set)
Let DSet be a category of digital sets, that is, DSet has digital images as objects and digital functions as morphisms. A digital simplicial set is a contravariant functor (equivalently, a covariant functor ).
4 Simplicial groups in digital images
Group axioms are satisfied in a finite subset of , including the point.
Definition 4.1 Let be a subset of the digital image which has κ-adjacency relation. A simplicial group in digital images consists of a sequence of groups and collections of group homomorphisms and , , that satisfy the following axioms:
An example of a simplicial group in a digital image can be seen in Figure 11.
Example 4.2 Suppose X is a digital simplicial set with κ-adjacency. Then we may define the simplicial group having as a free abelian group generated by the elements of with and in taken to be linear extensions of the face maps and of X.
We can also present the total face map and then define the homology as the homology of the chain complex .
Example 4.3 Let us give a remarkable example for a simplicial group in digital images, which is important in the theory of homology of groups in digital images. Assume that G is a group of digital images and DG is the simplicial group defined as follows. Let be the product of G with itself n times. Thus is just the trivial group . For an element , let
This defines a simplicial group in digital images. The realization of an underlying digital simplicial set turns out to be the classifying space of the group G, and so the homology coincides with group homology of the group G.
5 Digital homology groups of simplicial groups
In this section, we consider X as a digital simplicial group, and we use the method of calculations given by  and .
Definition 5.1 is a free abelian group with basis of all digital -simplices in X.
Corollary 5.2LetXbe anm-dimensional digital simplicial group withκ-adjacency. Then if , thenis a trivial group.
Definition 5.3 Let be an m-dimensional digital simplicial group. The homomorphism defined by
is called boundary homomorphism. (Here denotes the deletion of the point .)
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