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The digital smash product

  • In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.

    Citation: Ismet Cinar, Ozgur Ege, Ismet Karaca. The digital smash product[J]. Electronic Research Archive, 2020, 28(1): 459-469. doi: 10.3934/era.2020026

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  • In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.



    Digital topology with interesting applications has been a popular topic in computer science and mathematics for several decades. Many researchers such as Rosenfeld [21,22], Kong [18,17], Kopperman [19], Boxer, Herman [14], Kovalevsky [20], Bertrand and Malgouyres would like to obtain some information about digital objects using topology and algebraic topology.

    The first study in this area was done by Rosenfeld [21] at the end of 1970s. He introduced the concept of continuity of a function from a digital image to another digital image. Later Boxer [1] presents a continuous function, a retraction, and a homotopy from the digital viewpoint. Boxer et al. [7] calculate the simplicial homology groups of some special digital surfaces and compute their Euler characteristics.

    Ege and Karaca [9] introduce the universal coefficient theorem and the Eilenberg-Steenrod axioms for digital simplicial homology groups. They also obtain some results on the Künneth formula and the Hurewicz theorem in digital images. Ege and Karaca [10] investigate the digital simplicial cohomology groups and especially define the cup product. For other significant studies, see [13,12,16].

    Karaca and Cinar [15] construct the digital singular cohomology groups of the digital images equipped with Khalimsky topology. Then they examine the Eilenberg- Steenrod axioms, the universal coefficient theorem, and the Künneth formula for a cohomology theory. They also introduce a cup product and give general properties of this new operation. Cinar and Karaca [8] calculate the digital homology groups of various digital surfaces and give some results related to Euler characteristics for some digital connected surfaces.

    This paper is organized as follows: First, some information about the digital topology is given in the section of preliminaries. In the next section, we define the smash product for digital images. Then, we show that this product has some properties such as associativity, distributivity, and commutativity. Finally, we investigate a suspension and a cone for any digital image and give some examples.

    Let Zn be the set of lattice points in the n-dimensional Euclidean space. We call that (X,κ) is a digital image where X is a finite subset of Zn and κ is an adjacency relation for the members of X. Adjacency relations on Zn are defined as follows: Two points p=(p1,p2,,pn) and q=(q1,q2,,qn) in Zn are called cl-adjacent [2] for 1ln if there are at most l indices i such that |piqi|=1 and for all other indices i such that |piqi|1, pi=qi. It is easy to see that c1=2 (see Figure 1) in Z,

    Figure 1.  2-adjacency in Z.

    c1=4 and c2=8 (see Figure 2) in Z2,

    Figure 2.  4 and 8 adjacencies in Z2.

    and c1=6, c2=18 and c3=26 (see Figure 3) in Z3.

    Figure 3.  6, 18 and 26 adjacencies in Z3.

    A κ-neighbor of p in Zn is a point of Zn which is κ-adjacent to p. A digital image X is κ-connected [14] if and only if for each distinct points x,yX, there exists a set {a0,a1,,ar} of points of X such that x=a0, y=ar, and ai and ai+1 are κ-adjacent where i{0,1,,r1}. A κ-component of a digital image X is a maximal κ-connected subset of X. Let a,bZ with a<b. A digital interval [1] is defined as follows:

    [x,y]Z={aZ | xay,x,yZ},

    where 2-adjacency relation is assumed.

    In a digital image (X,κ), a digital κ-path [3] from x to y is a (2,κ)-continuous function f:[0,m]ZX such that f(0)=x and f(m)=y where x,yX. Let f:(X,κ)(Y,λ) be a function. If the image under f of every κ-connected subset of X is κ-connected, then f is called (κ,λ)-continuous [2].

    A function f:(X,κ)(Y,λ) is (κ,λ)-continuous [22,2] if and only if for any κ-adjacent points a,bX, the points f(a) and f(b) are equal or λ-adjacent. A function f:(X,κ)(Y,λ) is an isomorphism [4] if f is a (κ,λ)-continuous bijection and f1 is (λ,κ)-continuous.

    Definition 2.1. [2] Suppose that f, g:(X,κ)(Y,λ) are (κ,λ)-continuous maps. If there exist a positive integer m and a function

    F:X×[0,m]ZY

    with the following conditions, then F is called a digital (κ,λ)-homotopy between f and g, and we say that f and g are digitally (κ,λ)-homotopic in Y, denoted by f(κ,λ)g.

    (ⅰ) For all xX, F(x,0)=f(x) and F(x,m)=g(x).

    (ⅱ) For all xX, Fx:[0,m]Y defined by Fx(t)=F(x,t) is (2,λ)-continuous.

    (ⅲ) For all t[0,m]Z, Ft:XY defined by Ft(x)=F(x,t) is (κ,λ)-continuous.

    A digital image (X,κ) is κ-contractible [1] if the identity map on X is (κ,κ)-homotopic to a constant map on X.

    A (κ,λ)-continuous map f:XY is (κ,λ)-homotopy equivalence [3] if there exists a (λ,κ)-continuous map g:YX such that

    gf(κ,κ)1X   and   fg(λ,λ)1Y

    where 1X and 1Y are the identity maps on X and Y, respectively. Moreover, we say that X and Y have the same (κ,λ)-homotopy type.

    For the cartesian product of two digital images X1 and X2, the adjacency relation [6] is defined as follows: Two points xi,yi(Xi,κi), (x0,y0) and (x1,y1) are k(κ1,κ2)-adjacent in X1×X2 if and only if one of the following is satisfied:

    x0=x1 and y0=y1; or

    x0=x1 and y0 and y1 are κ1-adjacent; or

    x0 and x1 are κ0-adjacent and y0=y1; or

    x0 and x1 are κ0-adjacent and y0 and y1 are κ1-adjacent.

    Definition 2.2. [3] A (κ,λ)-continuous surjection f:XY is (κ,λ)-shy if

    for each yY, f1({y}) is κ-connected, and

    for each y0,y1Y, if y0 and y1 are λ-adjacent, then f1({y0,y1}) is κ-connected.

    Theorem 2.3. [5] For a continuous surjection f(X,κ)(Y,λ), if f is an isomorphism, then f is shy. On the other hand, if f is shy and injective, then f is an isomorphism.

    The wedge of two digital images (X,κ) and (Y,λ), denoted by XY, is the union of the digital images (X,μ) and (Y,μ), where [4]

    X and Y have a single point p;

    If xX and yY are μ-adjacent, then either x=p or y=p;

    (X,μ) and (X,κ) are isomorphic; and

    (Y,μ) and (Y,λ) are isomorphic.

    Theorem 2.4. [5] Two continuous surjections

    f:(A,α)(C,γ)   and   g:(B,β)(D,δ)

    are shy maps if and only if f×g:(A×B,k(α,β))(C×D,k(γ,δ)) is a shy map.

    Sphere-like digital images is defined as follows [4]:

    Sn=[1,1]n+1Z{0n+1}Zn+1,

    where 0n is the origin point of Zn. For n=0 and n=1, the sphere-like digital images are shown in Figure 4.

    S0={c0=(1,0),c1=(1,0)},
    S1={c0=(1,0),c1=(1,1),c2=(0,1),c3=(1,1),c4=(1,0),c5=(1,1),
    c6=(0,1),c7=(1,1)}.
    Figure 4.  Digital 0sphere S0 and digital 1-sphere S1.

    In this section, we define the digital smash product which has some important relations with a digital homotopy theory.

    Definition 3.1. Let (X,κ) and (Y,λ) be two digital images. The digital smash product XY is defined to be the quotient digital image (X×Y)/(XY) with the adjacency relation k(κ,λ), where XY is regarded as a subset of X×Y.

    Before giving some properties of the digital smash product, we prove some theorems which will be used later.

    Theorem 3.2. Let Xa and Ya be digital images for each element a of an index set A. For each aA, if fa(κ,λ)ga:XaYa then

    aAfa(κn,λn)aAga,

    where n is the cardinality of the set A.

    Proof. Let Fa:Xa×[0,m]ZYa be a digital (κ,λ)-homotopy between fa and ga, where [0,m]Z is a digital interval. Then

    F:(aAXa)×[0,m]ZaAYa

    defined by

    F((xa),t)=(Fa(xa,t))

    is a digital continuous function, where t is an element of [0,m]Z since the functions Fa are digital continuous for each element aA. Therefore F is a digital (κn,λn)-homotopy between aAfa and aAga.

    Theorem 3.3. If each fa:XaYa is a digital (κ,λ)-homotopy equivalence for all aA, then aAfa is a digital (κn,λn)-homotopy equivalence, where n is the cardinality of the set A.

    Proof. Let ga:YaXa be a (λ,κ)-homotopy inverse to fa, for each aA. Then we obtain the following relations:

    (aAga)(aAfa)=aA(ga×fa)(λn,κn)aA(1Xa)=1aAXa,
    (aAfa)(aAga)=aA(fa×ga)(κn,λn)aA(1Ya)=1aAYa.

    So we conclude that aAfa is a digital (κn,λn)-homotopy equivalence.

    Theorem 3.4. Let (X,κ), (Y,λ) and (Z,σ) be digital images. If p:(X,κ)(Y,λ) is a (κ,λ)shy map and (Z,σ) is a σ-connected digital image, then

    p×1:(X×Z,k(κ×σ))(Y×Z,k(λ×σ))

    is a (κ×σ,λ×σ)-shy map, where 1Z:(Z,σ)(Z,σ) is an identity function.

    Proof. Since (Z,σ) is a σ-connected digital image, then for yY and zZ, we have

    (p×1Z)1(y,z)=(p1(y),11Z(z))=(p1(y),z).

    Thus, for each yY and zZ, (p×1Z)1(y,z) is κ-connected by the definition of the adjacency of the cartesian product of digital images. Moreover, the map 1Z preserves the connectivity, that is, for every z0,z1Z such that z0 and z1 are σ-adjacent, 1Z({z0,z1})={z0,z1} is σ-connected. It is easy to see that

    (p×1Z)1({y0,y1},{z0,z1})=(p1({y0,y1}),11Z({z0,z1}))=(p1({y0,y1}),({z0,z1})).

    Hence for each y0,y1Y and z0,z1Z, (p×1Z)1({y0,y1},{z0,z1}) is a k(κ,σ)-connected using the definition of the adjacency of the Cartesian product of digital images.

    Theorem 3.5. Let A and B be digital subsets of (X,κ) and (Y,λ), respectively. If f,g:(X,A)(Y,B) are (κ,λ)-continuous functions such that f(κ,λ)g, then the induced maps ˉf,ˉg:(X/A,κ)(Y/B,λ) are digitally (κ,λ)-homotopic.

    Proof. Let F:(X×I,A×I)(Y,B) be a digital (κ,λ)-homotopy between f and g where I=[0,m]Z. It is clear that F induces a digital function ˉF:(X/A)×IY/B such that the following square diagram is commutative, where p and q are shy maps:

    (p×1Z)1({y0,y1},{z0,z1})=(p1({y0,y1}),11Z({z0,z1}))=(p1({y0,y1}),({z0,z1})).

    Since qF is digitally continuous, p×1 is a shy map and ˉF(p×1)=qF, ˉF is a digital continuous map. Hence ˉF is a digital (κ,λ)-homotopy map between ˉf and ˉg.

    We are ready to present some properties of the digital smash product. The following theorem gives a relation between the digital smash product and the digital homotopy.

    Theorem 3.6. Given digital images (X,κ), (Y,λ), (A,σ), (B,α) and two digital functions f:XA and g:YB, there exists a function fg:XYAB with the following properties:

    (i) If h:AC, k:BD are digital functions, then

    (hk)(fg)=(hf)(kg).

    (ii) If f(κ,σ)f:XA and g(λ,α)g:YB, then

    fg(k(κ,λ),k(σ,α))fg.

    Proof. The digital function f×g:X×YA×B has the property that

    (f×g)(XY)A×B.

    Hence f×g induces a digital function fg:XYAB and property (i) is obvious. As for (ii), the digital homotopy F between f×g and f×g can be constructed as follows: We know that

    f(κ,σ)f   and   g(λ,α)g.

    By Theorem 3.2, we have

    f×g(k(κ,λ),k(σ,α))f×g.

    F is a digital homotopy of functions of pairs from (X×Y,XY) to (A×B,AB). Consequently a digital homotopy between fg and fg is induced by Theorem 3.5.

    Theorem 3.7. If f and g are digital homotopy equivalences, then fg is a digital homotopy equivalence.

    Proof. Let f:(X,κ)(Y,λ) be a (κ,λ)-homotopy equivalence. Then there exists a (λ,κ)-continuous function f:(Y,λ)(X,κ) such that

    ff(λ,λ)1Y and ff(κ,κ)1X.

    Moreover, let g:(A,σ)(B,α) be a (σ,α)-homotopy equivalence. Then there is a (α,σ)-continuous function g:(B,α)(A,σ) such that

    gg(α,α)1B and gg(σ,σ)1A.

    By Theorem 3.6, there exist digital functions

    fg:XAYB   and   fg:YBXA

    such that

    (fg)(fg)=1YB,
    (ff)(gg)=1YB,

    and

    (fg)(fg)=1XA,
    (ff)(gg)=1XA.

    So fg is a digital homotopy equivalence.

    The following theorem shows that the digital smash product is associative.

    Theorem 3.8. Let (X,κ), (Y,λ) and (Z,σ) be digital images. (XY)Z is digitally isomorphic to X(YZ).

    Proof. Consider the following diagram:

    (ff)(gg)=1XA.

    where p represents for the digital shy maps of the form X×YXY. By Theorem 3.4, p×1 and 1×p are digital shy maps. 1:X×Y×ZX×Y×Z induces functions

    f:(XY)ZX(YZ)   and   g:X(YZ)(XY)Z.

    These functions are clearly injections. By Theorem 2.3, f is a digital isomorphism.

    The next theorem gives the distributivity property for the digital smash product.

    Theorem 3.9. Let (X,κ), (Y,λ) and (Z,σ) be digital images. (XY)Z is digitally isomorphic to (XZ)(YZ).

    Proof. Suppose that p represents for the digital shy maps of the form X×YXY and q stands for the digital shy maps of the form X×YXY. We may obtain the following diagram:

    f:(XY)ZX(YZ)   and   g:X(YZ)(XY)Z.

    From Theorem 2.4, p×p is a digital shy map and by Theorem 3.4, q1 is also a digital shy map. The function m:(X×Y)×Z(X×Z)×(Y×Z) induces a digital function

    f:(XZ)×(YZ)(X×Z)×(Y×Z).

    Obviously f is a one-to-one function. By Theorem 2.3, f is a digital isomorphism.

    Theorem 3.10. Let (X,κ) and (Y,λ) be digital images. XY is digitally isomorphic to YX.

    Proof. If we suppose that g stands for the digital shy maps Y×XYX and p represents for the digital shy maps of the form X×YXY, we get the following diagram:

    f:(XZ)×(YZ)(X×Z)×(Y×Z).

    The switching map u:X×YY×X induces a digital shy map f:XYYX. Additionally, f is a one-to-one. Hence, f is a digital isomorphism from Theorem 2.3.

    Definition 3.11. The digital suspension of a digital image X, denoted by sX, is defined to be XS1.

    Example 1. Choose a digital image X=S0. Then we get the following digital images in Figure 5.

    Figure 5.  S1×S0 and S1S0.

    Theorem 3.12. Let x0 be the base point of a digital image X. Then sX is digitally isomorphic to the quotient digital image

    (X×[a,b]Z)/(X×{a}{x0}×[a,b]X×{b}),

    where the cardinality of [a,b]Z is equal to 8.

    Proof. The function

    [a,b]ZθS1

    is a digital shy map defined by θ(ti)=ci mod 8, where ciS1 and i{0,1,,7}. Hence if p:X×S1XS1 is a digital shy map, then the digital function

    X×[a,b]Z1×θX×S1pXS1

    is also a digital shy map, and its effect is to identify together points of

    X×{a}{x0}×[a,b]ZX×{b}.

    The digital composite function p(1×θ) induces a digital isomorphism

    (X×[a,b]Z)/(X×{a}{x0}×[a,b]ZX×{b})XS1=sX.

    Definition 3.13. The digital cone of a digital image X, denoted by cX, is defined to be XI, where I=[0,1]Z.

    Example 2. Take a digital image X=S0. Then we have the following digital images in Figure 6.

    Figure 6.  S0×I and S0I.

    Theorem 3.14. For any digital image (X,κ), the digital cone cX is a contractible digital image.

    Proof. Since I=[0,1]Z is digitally contractible to the point {0},

    cX=XI(2,2)X{0}

    is obviously a single point.

    Corollary 1. For mN, SmI is equal to SmS0, where I=[0,1]Z is the digital interval and S0 is a digital 0-sphere.

    Proof. Since S0 and I consist of two points, we get the required result.

    For each m,n0, can we prove that digital (m+n)-sphere Sm+n is isomorphic to SmSn?

    This paper introduces some notions such as the smash product, the suspension, and the cone for digital images. Since they are significant topics related to homotopy, homology, and cohomology groups in algebraic topology, we believe that the results in the paper can be useful for future studies in digital topology.

    We would like to express our gratitude to the anonymous referees for their helpful suggestions and corrections.



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