2. Preliminaries
Let
be the set of all integers, and let:
be the set of all functions:
such that:
for each
; that is, the set of all lattice points
in the
n-dimensional Euclidean space
for
, where
for
. We note that the above function:
can be identified with the lattice point
by defining:
to be:
for each
.
For a positive integer u with , we define an adjacency relation as follows.
Definition 1 ([
26]).
The two different functions or the two different points and in are said to be -adjacent if:there are at most u distinct indices i with the property ; and
for all indices j, if , then .
Example 1. - (1)
The set of -adjacent points of zero in is the set consisting of and one.
- (2)
The set of -adjacent points of in is the set consisting of , , , and .
- (3)
The set of -adjacent points of in is the set consisting of , , , , , , , and .
A -adjacency relation on may be considered as the cardinality of the set of lattice points that have the -adjacency relations centered at in . From this point of view, we define the following.
Definition 2. The -adjacent points of are said to be two-adjacent.
Convention: We sometimes denote the -adjacency relation on a digital image X by the -adjacency relation for short unless we specifically state otherwise.
A digital image
consists of a bounded and finite subset
X of
and an adjacency relation
on
X. A digital image
in
is said to be
-connected ([
27,
28]) if for each set
consisting of two distinct points
x and
y, there exists a subset:
consisting of
distinct points such that:
Example 2. Let be a subset of , where:
;
;
and
.
Then, X is a digital image with eight-adjacent relation in ; see Figure 1. Definition 3 ([
29]).
Let and be the digital images with -adjacency and -adjacency relations, respectively. A function:of digital images is said to be a -continuous function if the image of every -connected subset of the digital image under f is a -connected subset of ; see also ([28] (Definition 2.3)). Let
a and
b be elements of
, considered as a simply ordered set, with
. The bounded and finite set:
considered as a digital image with the two-adjacency relation in
is said to be a digital interval; see [
27,
30]. A triple
consisting of a digital image
X with a
-adjacency relation and an element
of
X is said to be a pointed digital image [
31]. Here,
is called a base point of
just like a base point of a pointed topological space. A
-continuous function of pointed digital images:
such that:
is called a base point preserving digital continuous function (or a pointed digital continuous function).
Definition 4 ([
26,
28,
32]).
Let and be pointed digital images with -adjacency and -adjacency relations, respectively, and let:be the base point preserving -continuous functions. Assume that there is a function:such that: and for all ;
the function defined by for all is -continuous;
the function defined by for all is -continuous; and
for all ,
where is a digital interval. Then, F is called a pointed digital -homotopy between f and g, usually written as:and f and g are called pointed digital -homotopic in Y. As usual, we denote the pointed digital homotopy class by as the equivalence class of a base point preserving -continuous function .
Remark 1. We note that if , , and are pointed digital images and if is a -continuous function and is a -continuous function, then it can be shown that the composite is also a -continuous function. Therefore, it is possible for us to consider the pointed digital category whose object classes are pointed digital images and whose morphism classes are base point preserving digital continuous functions. By ignoring the base points once in a while, we can also construct the category whose object classes are digital images and whose morphism classes are digital continuous functions.
3. Digital Homology and Cohomology Modules
A module is one of the pivotal algebraic structures in algebra and algebraic topology. A module over a ring is a generalization of the notion of a vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity).
Historically, classical cohomology modules were not defined until long after homology modules because cohomology modules are much less natural than homology modules geometrically. The mathematical term “homology” was first used in a topological sense by H. Poincaré in 1897.
In this section, we explore the digital homology and cohomology modules of digital images and construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms, where R is a commutative ring with identity .
Let
,
, …, and
be the elements of
, and let
be a finite subset of
consisting of all of these elements; that is,
Similarly, as described earlier, we denote the -adjacency relation in the digital image as a finite subset of by for our notational convenience.
For a digital image
with a
-adjacency relation, a
-continuous function:
is called a digital
n-simplex in
. We note that for every
so that the set
is a
-connected subset of
, either:
or
is a
-connected subset of
.
Let
R be a commutative ring with identity
, and let
be a digital image with a
-adjacency relation. For each
, we denote
by the non-negatively graded free
R-module with the basis consisting of all digital
n-simplexes
in
and define the so-called digital boundary operator:
by
where
is the
i-th face function; see [
33,
34]. It can be shown that:
for all
, and thus,
is automatically an
R-submodule of
for each
. The
n-th digital homology module
over
R of a digital image
with a
-adjacency relation is defined by:
for each
[
33]; see also [
2,
35].
We observe that if the commutative ring R with identity is equal to the ring of integers, then the non-negatively graded free -module (or free Abelian group) is denoted by for short, and similarly for the n-th digital homology module.
We now explore the digital cohomology modules over a commutative ring R with identity as follows.
Definition 5. The module of n-cochains of a digital image , with coefficients in a commutative ring R with identity , is defined to be the R-module:whose scalar multiplication:is given by:where: Definition 6. The digital coboundary operator is defined as the dual of the digital boundary operator that is,for each . Remark 2. We note that if R is a commutative ring with identity , then is a unitary R-module. Indeed, we have:for all ; that is,for all . Definition 7. The kernel of:is said to be the module of digital n-cocycles in and is denoted by for all . Definition 8. The image of:is said to be the module of digital n-coboundaries in and is denoted by for all . From Definitions 7 and 8, we can see that:
and are R-submodules of ;
for all ; and
is an R-submodule of .
Therefore, we can define the following.
Definition 9. The n-th digital cohomology module over R of a digital image is defined to be the corresponding cohomology module over R of the cochain complex obtained by the dual R-modules along with the dual R-module homomorphisms, i.e.,for all . Indeed, as a quotient R-module, has the unitary R-module structure because R is a commutative ring with identity; see below.
Remark 3. If K is a simplicial complex, then the classical simplicial homology of K is isomorphic to the singular homology of the polytope of ([36] (Theorem 34.3)), and similarly for cohomology groups from the universal coefficient theorem for cohomology. In the context of digital images, on the other hand, there is a small difference between the n-th simplicial cohomology group ([4] (Definition 3.2)) and the n-th digital cohomology module in that the n-th digital cohomology module does not require dividing a digital image into n-simplexes. However, it is well known in mathematics that the smooth compact surfaces can be triangulated. Let
A and
B be
R-modules. A function
is said to be an
R-module homomorphism if the following diagrams:
and:
are strictly commutative, where:
and are the binary operations on A and B, respectively;
the bullets • in blue are the scalar multiplications on the R-modules A and B with the same notation; and
is the identity function on the commutative ring R with identity.
We note that if:
is a digital
n-simplex in
and if:
is a
-continuous function, then it can be shown that:
is a digital
n-simplex in
. Therefore, by using the linear property, we have an
R-module homomorphism of
R-modules:
defined by:
where
is an element of the commutative ring
R with identity, and the bullets
• in blue are the scalar multiplications on the
R-modules
and
with the same notation.
Moreover, we have the following.
Remark 4. The following diagram:strictly commutes for every , where is the digital boundary operator of a digital image . Let
and
be categories. We recall that a contravariant functor:
consists of an object function, which assigns to every object
X of
an object
of
and a morphism function that assigns to every morphism
of
a morphism
of
such that:
and:
where:
is the identity morphism in ;
is the identity morphism in ;
is a morphism class in ; and
is a morphism class in .
As usual, we define a map:
by:
for every
; that is, the following triangle:
is strictly commutative, where
is the ring of integers.
For any element
r of
R and any element
of
, it can be seen that the map:
preserves the addition and the scalar multiplication. Indeed, we have:
for all
, where the bullets
• in blue are the scalar multiplications on the
R-module structures of
or
. Therefore, we have the following.
Lemma 1. Let R be a commutative ring with identity . Then, any -continuous function:induces an R-module homomorphism of unitary R-modules: Proof. See ([
37] (Proposition 2.5)) for more details. □
We note that the following diagram:
is strictly commutative, where
and
are digital coboundary operators of the digital images
and
, respectively, for all
.
It can be seen that, for each digital image
,
is a unitary
R-module whose scalar multiplication:
is given by:
where:
Indeed, by Remark 2, we have:
for all
, where
is the (multiplicative) identity in
R.
Lemma 2. Let be a -continuous function. Then, the map:given by:is an R-module homomorphism, where with . Proof. If
is any element of
, then:
and thus:
that is,
Therefore, we have:
for each
. Similarly, if
is any element of
, then:
for some
, and:
Therefore, we obtain:
for each
.
If
is an element of
, then:
that is, the definition of
is independent of the choice of representatives. Thus, the map:
is well defined.
We now show that:
is an
R-module homomorphism for each
. Let:
and:
be the elements of
. Then, we have:
that is,
preserves the addition.
We note that
has the
R-module structure for each digital image
and that:
is an
R-module homomorphism by Lemma 1; that is, the following diagram:
is strictly commutative. Therefore, for all
and
, we also obtain:
where ⋄ is the ring multiplication in the commutative ring
R with identity
considered as the
R-module over itself as in Definition 5; that is,
preserves the scalar multiplication, as required. □
Let be the category of digital images and digital continuous functions as mentioned earlier in Remark 1, and let be the category of unitary R-modules and R-module homomorphisms. Then, we have the following theorem.
Theorem 1. Let R be a commutative ring with identity . Then, the functor:given by:is a contravariant functor for each . Proof. If
is the identity morphism in
and if
is any digital
n-cocycle, i.e.,
then we have:
that is,
, the identity automorphism on
for each
.
If
is a digitally
-continuous function and
is a digitally
-continuous function, then, by Lemmas 1 and 2, we obtain:
for any
,
. Therefore, we have:
for each
, as required. □
Remark 5. The above results assert that for a digital image X, the assignment is a contravariant functor F from the category of digital images and digital continuous functions to the category of cochain complexes of R-modules and cochain maps. Similarly, the assignment is a covariant functor ; see ([37] (Proposition 6.8)). We note that the following triangle:is commutative. 4. Digital Primitive Cohomology Classes
In algebra, a primitive element of a co-algebra
C over an element
g is an element
x that satisfies:
where
is the so-called algebraic comultiplication and
g is an element of
C that maps to the multiplicative identity
of the base field (or commutative ring with identity
) under the counit; see ([
38] (page 510)).
In this section, we define a digital primitive cohomology class and find out the relationship between
R-module homomorphisms of digital cohomology
R-modules induced by the digital convolutions and digital continuous functions based on the digital Hopf spaces with digital homotopy multiplications (compare with [
39,
40]) as the immediate application of a Hopf space in algebraic topology; see also [
8,
41].
Let
be a constant function at
, and let:
be the identity function on
.
Definition 10 ([
31,
42]).
A pointed digital Hopf space consists of a pointed digital image with an adjacency relation and a -continuous function , which is called a digital homotopy multiplication (or digital multiplication for short) such that the following diagrams:and:commutate up to pointed digital homotopy, where is the diagonal function. Example 3 ([
42], Example 3.8).
Let be a digital image with the eight-adjacent relation, where and as in Example 2. If we define a binary operation by the rule of Table 1, then it can be shown that the pointed digital image is a digital Hopf space with the digital multiplication:where and and is digital homotopy associative and commutative. There exists also a digital homotopy inverse:so that becomes a digital commutative Hopf group. Indeed, it is a pointed digital Hopf space. One of the reasons for considering a Hopf space with a digital homotopy multiplication is to define the digital version of the usual convolution in mathematics as follows.
Definition 11 ([
31,
42]).
If is a digital Hopf space with a digital homotopy multiplication and if:are the base point preserving -continuous functions, then a digital convolution of and is defined by the digital homotopy class of the composition:that is,where is the diagonal function. Let be the first and second projections onto Y, respectively.
Definition 12. Let be a digital Hopf space with a digital homotopy multiplication . Then, an element is said to be a digital primitive cohomology class if:where:are the R-module homomorphisms of digital cohomology modules induced by the first and second projections respectively; see ([38] (page 804)) and ([8] (page 143)) in the sense of coalgebra and homotopy theory, respectively. Definition 13 ([
31,
34]).
Let and be pointed digital Hopf spaces with digital multiplications and , respectively. A base point preserving -continuous function:is said to be a digital Hopf function if and are pointed digital -homotopic in ; that is, Remark 6. One of the Eilenberg–Steenrod axioms is the homotopy axiom in (generalized) homology and cohomology theories, K-theory, bordism and cobordism theories, Brown–Peterson cohomology theory, and so on; that is, they are invariant under homotopy equivalences. More precisely, homotopic maps induce the same morphism in those theories, i.e., if , then , where C is the covariant (or contravariant) functor inducing those theories (compare with [4]). In our case, it can be seen that if:then: Indeed, if:
is a digital
-homotopy between
f and
g; that is,
, and if:
is a
-continuous function given by:
for all
, then there exists an
R-module homomorphism:
such that:
that is,
plays the role of a chain homotopy between
and
, where:
is a digital boundary operator; and
are R-module homomorphisms (as chain maps) induced by and , respectively.
Since:
and:
we have:
at the level of digital homology modules. By applying the contravariant functor
to the above chain homotopy with a little more work, we have the result of Remark 6; see ([
43] (Theorems 4.23 and 12.4)) for more details.
Let be the R-submodule of consisting of all the digital primitive cohomology classes. Then, we have the following theorem.
Theorem 2. Let and be digital Hopf spaces with digital multiplications and , respectively, and let be a digital Hopf function of digital Hopf spaces. Then, we obtain:where:is an R-module homomorphism of digital cohomology modules induced by the -continuous functions f. Proof. Let
be the first and second projections, respectively, onto
W, where
or
Y. Then, by the contravariant functorial property of the digital cohomology
R-modules in Theorem 1, the following commutative diagram:
induces a commutative diagram:
for each
.
Since
is a digital Hopf function, by using the digital homotopy relation (
2) and Remark 6, we see that the following diagram:
is also strictly commutative.
If
y is any digital primitive cohomology class in the digital cohomology module
with coefficients in a commutative ring
R with identity, that is
, then we obtain:
Therefore, the image of the
R-submodule
of
under the
R-module homomorphism:
is also an
R-submodule
of
, as required. □
We now find the relationship between R-module homomorphisms of digital cohomology R-modules induced by the digital convolutions and digital continuous functions.
Theorem 3. Let be a digital Hopf space with a digital multiplication , and let:be -continuous functions. If is any digital primitive cohomology class; that is, , then:where:are R-module homomorphisms of digital cohomology modules induced by the -continuous functions f and g, respectively, and similarly for . Proof. Once again, we let
be the first and second projections, respectively, onto
W, where
or
Y. The contravariant functorial property in Theorem 1 asserts that the following commutative triangle:
induces a strictly commutative diagram of digital cohomology
R-modules:
for each
, where:
is the diagonal function;
is the identity function; and
is the identity automorphism on .
Similarly, the commutative rectangles:
and:
induce the strictly commutative diagrams of digital cohomology
R-modules:
and:
respectively. The digital convolution of
and
says that:
Therefore, if
, then we have:
for all
and
. We note that the third equality above is guaranteed by the fact that
y is a digital primitive cohomology class, and the fifth equality comes from the
R-module homomorphism:
induced by the diagonal function
, as required. □
Remark 7. The hypothesis of the digital primitive cohomology class in Theorem 3 is absolutely necessary. In general, if y is an element of , which is not necessarily digital primitive, and if:where is the sum of decomposable digital cohomology classes, then it can be seen that: 5. Conclusions and Further Prospects
Hopf spaces and their Eckmann–Hilton dual notions play an important role in classical (equivariant) homotopy theory in algebraic topology. The digital topology concerns features and properties of the digital images in , especially the two-dimensional or three-dimensional digital images corresponding to the topological features and properties of objects.
In this paper, we investigated some fundamental properties of the digital cohomology modules and the primitive cohomology classes of digital images. More specifically, this study focused on constructing the contravariant functor from the category of (pointed) digital images and (base point preserving) digital continuous functions to the category of unitary R-modules and R-module homomorphisms, where R is a commutative ring with identity . By using the contravariant functorial property in digital cohomology modules, we also examined the digital primitive cohomology classes and developed the relationship between R-module homomorphisms of digital cohomology R-modules induced by the digital convolutions and digital continuous functions.
We hope that our methods will be used to study the advantages of coalgebras along with the Hopf algebras, which will be studied in the near future as a subsequent paper. We also hope that the results will be applied to the concepts of various (algebra) comultiplications with many kinds of perturbations on the (algebraic) objects in many areas of algebra, the algebraic topology, and computer sciences.