
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
and
A
[x,y]Z={a∈Z | x≤a≤y,x,y∈Z}, |
where
In a digital image
A function
Definition 2.1. [2]
Suppose that
F:X×[0,m]Z→Y |
with the following conditions, then
(ⅰ) For all
(ⅱ) For all
(ⅲ) For all
A digital image
A
g∘f≃(κ,κ)1X and f∘g≃(λ,λ)1Y |
where
For the cartesian product of two digital images
Definition 2.2. [3]
A
Theorem 2.3. [5] For a continuous surjection
The wedge of two digital images
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
Sphere-like digital images is defined as follows [4]:
Sn=[−1,1]n+1Z∖{0n+1}⊂Zn+1, |
where
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)}. |
In this section, we define the digital smash product which has some important relations with a digital homotopy theory.
Definition 3.1. Let
Before giving some properties of the digital smash product, we prove some theorems which will be used later.
Theorem 3.2.
Let
∏a∈Afa≃(κn,λn)∏a∈Aga, |
where
Proof. Let
F:(∏a∈AXa)×[0,m]Z→∏a∈AYa |
defined by
F((xa),t)=(Fa(xa,t)) |
is a digital continuous function, where
Theorem 3.3. If each
Proof. Let
(∏a∈Aga)(∏a∈Afa)=∏a∈A(ga×fa)≃(λn,κn)∏a∈A(1Xa)=1∏a∈AXa, |
(∏a∈Afa)(∏a∈Aga)=∏a∈A(fa×ga)≃(κn,λn)∏a∈A(1Ya)=1∏a∈AYa. |
So we conclude that
Theorem 3.4.
Let
p×1:(X×Z,k∗(κ×σ))→(Y×Z,k∗(λ×σ)) |
is a
Proof. Since
(p×1Z)−1(y,z)=(p−1(y),1−1Z(z))=(p−1(y),z). |
Thus, for each
(p×1Z)−1({y0,y1},{z0,z1})=(p−1({y0,y1}),1−1Z({z0,z1}))=(p−1({y0,y1}),({z0,z1})). |
Hence for each
Theorem 3.5.
Let
Proof. Let
(p×1Z)−1({y0,y1},{z0,z1})=(p−1({y0,y1}),1−1Z({z0,z1}))=(p−1({y0,y1}),({z0,z1})). |
Since
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
(h∧k)∘(f∧g)=(h∘f)∧(k∘g). |
f∧g≃(k∗(κ,λ),k∗(σ,α))f′∧g′. |
Proof. The digital function
(f×g)(X∨Y)⊂A×B. |
Hence
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′. |
Theorem 3.7.
If
Proof. Let
f∘f′≃(λ,λ)1Y and f′∘f≃(κ,κ)1X. |
Moreover, let
g∘g′≃(α,α)1B and g′∘g≃(σ,σ)1A. |
By Theorem 3.6, there exist digital functions
f∧g:X∧A→Y∧B and f′∧g′:Y∧B→X∧A |
such that
(f∧g)∘(f′∧g′)=1Y∧B, |
(f∘f′)∧(g∘g′)=1Y∧B, |
and
(f′∧g′)∘(f∧g)=1X∧A, |
(f′∘f)∧(g′∘g)=1X∧A. |
So
The following theorem shows that the digital smash product is associative.
Theorem 3.8.
Let
Proof. Consider the following diagram:
(f′∘f)∧(g′∘g)=1X∧A. |
where
f:(X∧Y)∧Z→X∧(Y∧Z) and g:X∧(Y∧Z)→(X∧Y)∧Z. |
These functions are clearly injections. By Theorem 2.3,
The next theorem gives the distributivity property for the digital smash product.
Theorem 3.9.
Let
Proof. Suppose that
f:(X∧Y)∧Z→X∧(Y∧Z) and g:X∧(Y∧Z)→(X∧Y)∧Z. |
From Theorem 2.4,
f:(X∧Z)×(Y∧Z)→(X×Z)×(Y×Z). |
Obviously
Theorem 3.10.
Let
Proof. If we suppose that
f:(X∧Z)×(Y∧Z)→(X×Z)×(Y×Z). |
The switching map
Definition 3.11. The digital suspension of a digital image
Example 1. Choose a digital image
Theorem 3.12. Let
(X×[a,b]Z)/(X×{a}∪{x0}×[a,b]∪X×{b}), |
where the cardinality of
Proof. The function
[a,b]Zθ⟶S1 |
is a digital shy map defined by
X×[a,b]Z1×θ⟶X×S1p⟶X∧S1 |
is also a digital shy map, and its effect is to identify together points of
X×{a}∪{x0}×[a,b]Z∪X×{b}. |
The digital composite function
(X×[a,b]Z)/(X×{a}∪{x0}×[a,b]Z∪X×{b})→X∧S1=sX. |
Definition 3.13. The digital cone of a digital image
Example 2. Take a digital image
Theorem 3.14. For any digital image
Proof. Since
cX=X∧I≃(2,2)X∧{0} |
is obviously a single point.
Corollary 1. For
Proof. Since
For each
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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