Computing Hypercrossed Complex Pairings in Digital Images

Abstract

We consider an additive group structure in digital images and introduce the commutator in digital images. Then we calculate the hypercrossed complex pairings which generates a normal subgroup in dimension 2 and in dimension 3 by using 8-adjacency and 26-adjacency.

1. Introduction

In this paper we denote the set of integers by ℤ. Then ℤ^𝑛 represents the set of lattice points in Euclidean 𝑛-dimensional spaces. A finite subset of ℤ^𝑛 with an adjacency relation is called a digital image.

Definition 1 (see [1, 2]). Consider the following.(1)Two points 𝑝 and 𝑞 in ℤ are 2-adjacent if |𝑝−𝑞|=1.(2)Two points 𝑝 and 𝑞 in ℤ² are 8-adjacent if they are distinct and differ by at most 1 in each coordinate.(3)Two points 𝑝 and 𝑞 in ℤ² are 4-adjacent if they are 8-adjacent and differ by exactly one coordinate.(4)Two points 𝑝 and 𝑞 in ℤ² are 26-adjacent if they are distinct and differ by at most 1 in each coordinate.(5)Two points 𝑝 and 𝑞 in ℤ² are 18-adjacent if they are 26-adjacent and differ in at most two coordinates.(6)Two points 𝑝 and 𝑞 in ℤ² are 6-adjacent if they are 18-adjacent and differ by exactly one coordinate.

Definition 2. Let 𝐆 be a subset of a digital image. A simplicial group 𝐆 in digital images consists of a sequence of groups 𝐆 and collections of group homomorphisms 𝑑_𝑖∶𝐺_𝑛→𝐺_𝑛−1 and 𝑠_𝑖∶𝐺_𝑛→𝐺_𝑛+1, 0≤𝑖≤𝑛, that satisfies the following axioms: 𝑑_𝑖𝑑_𝑗=𝑑_𝑗−1𝑑_𝑖𝑑,𝑖<𝑗,_𝑖𝑠_𝑗=𝑠_𝑗−1𝑑_𝑖𝑑,𝑖<𝑗,_𝑗𝑠_𝑗=𝑑_𝑗+1𝑠_𝑗𝑑=𝑖𝑑,𝑖=𝑗or𝑖=𝑗+1,_𝑖𝑠_𝑗=𝑠_𝑗𝑑_𝑖−1𝑠,𝑖>𝑗+1,_𝑖𝑠_𝑗=𝑠_𝑗+1𝑠_𝑖,𝑖≤𝑗. (1)

Definition 3. Given a simplicial group 𝐆 with 𝜅-adjacency, the Moore complex (𝐍𝐆,𝜕) of 𝐆 is the chain complex defined by 𝑁𝐺_𝑛=^𝑛−1_𝑖=0Ker𝑑_𝑖, (2) with 𝜕∶𝑁𝐺_𝑛→𝑁𝐺_𝑛−1 induced from 𝑑_𝑛 by restriction.
The 𝑛th homology group of the Moore complex of 𝐆 is 𝐻_𝑛()=⋂𝐍𝐆,𝜕^𝑛𝑖=0Ker𝑑_𝑖𝑑_𝑛+1⋂^𝑛_𝑖=0Ker𝑑_𝑖. (3)

2. Hypercrossed Complex Pairings in Digital Images

First of all we adapt ideas from Carrasco and Cegarra [3–5] to get the construction in digital images. We define a set 𝑃(𝑛) consisting of pairs of elements (𝛼,𝛽) from 𝑆(𝑛) with 𝛼∩𝛽=Ø and 𝛽<𝛼, with respect to lexicographic ordering in 𝑆(𝑛) where 𝛼=(𝑖_𝑙,…,𝑖₁) and 𝛽=(𝑗_𝑚,…,𝑗₁)∈𝑆(𝑛).

Consider the following diagram: (4) where 𝑠_𝛼=𝑠_𝑖𝑙⋯𝑠_𝑖1∶𝑁𝐺_𝑛−#𝛼⟶𝐺_𝑛,𝑠_𝛽=𝑠_𝑗𝑚⋯𝑠_𝑗1∶𝑁𝐺_𝑛−#𝛽⟶𝐺_𝑛, (5) and define 𝑝∶𝐺_𝑛→𝑁𝐺_𝑛 and 𝑝(𝑥)=𝑝_𝑛−1⋯𝑝₀(𝑥) as 𝑝_𝑗(𝑧)=𝑧−𝑠_𝑖𝑑_𝑖𝑧 and 𝑗=0,…,𝑛−1. Since a digital image has the additive group structure, define the commutator as 𝑥,𝑦=𝑥𝑦−𝑦𝑥. (6) Thus 𝜇∶𝐺_𝑛×𝐺_𝑛⟶𝐺_𝑛,𝐹_𝛼,𝛽𝑥_𝛼,𝑦_𝛽=𝑝𝜇𝑠_𝛼×𝑠_𝛽𝑥_𝛼,𝑦_𝛽=𝑝𝑠_𝛼𝑥_𝛼,𝑠_𝛽𝑦_𝛽. (7) The normal subgroup 𝑁𝐺_𝑛 of 𝐺_𝑛 is generated by the elements of the form 𝐹_𝛼,𝛽𝑥_𝛼,𝑦_𝛽, (8) where 𝑥_𝛼∈𝑁𝐺_𝑛−#𝛼 and 𝑦_𝛽∈𝑁𝐺_𝑛−#𝛽.

Theorem 4. 2-dimensional normal subgroup 𝑁₂ with 8-adjacency is generated by the elements of the form 𝑠₀𝑥₁−𝑠₁𝑥₁,𝑠₁𝑦₁. (9)

Proof. Let 𝛼=(1) and 𝛽=(0) for 𝑛=2. For 𝑥₁ and 𝑦₁∈𝑁𝐺₁=Ker𝑑₀, 𝐹_(0),(1)𝑥₁,𝑦₁=𝑝₁𝑝₀𝑠₀𝑥₁,𝑠₁𝑦₁=𝑝₁𝑠₀𝑥₁,𝑠₁𝑦₁−𝑠₀𝑑₀𝑠₀𝑥₁,𝑠₁𝑦₁=𝑠₀𝑥₁−𝑠₁𝑥₁,𝑠₁𝑦₁. (10) Thus 𝐹_(0),(1)(𝑥₁,𝑦₁)=[𝑠₀𝑥₁−𝑠₁𝑥₁,𝑠₁𝑦₁] and this is the element generating 𝑁₂ normal subgroups.

Proposition 5. 3-dimensional normal subgroup 𝑁₃ with 26-adjacency is generated by the elements of the following forms:(i)[𝑠₁𝑠₀𝑥₁−𝑠₀𝑠₁𝑥₁,𝑠₂𝑦₂],(ii)[𝑠₂𝑠₀𝑥₁−𝑠₂𝑠₁𝑥₁,𝑠₁𝑦₂−𝑠₂𝑦₂],(iii)[𝑠₀𝑥₂−𝑠₁𝑥₂+𝑠₂𝑥₂,𝑠₁𝑠₁𝑦₁],(iv)[𝑠₁𝑥₂−𝑠₂𝑥₂,𝑠₂𝑦₂],(v)[𝑠₀𝑥₂,𝑠₂𝑦₂],(vi)[𝑠₀𝑥₂−𝑠₁𝑥₂,𝑠₁𝑦₂]+[𝑠₂𝑥₂,𝑠₂𝑦₂].

Proof. For 𝑛=3 the possible pairings are the following:(i)𝐹_(1,0)(2),(ii)𝐹_(2,0)(1),(iii)𝐹_(0)(2,1),(iv)𝐹₍₁₎₍₂₎,(v)𝐹₍₀₎₍₂₎,(vi)𝐹₍₀₎₍₁₎.For all 𝑥₁∈𝑁𝐺₁ and 𝑦₂∈𝑁𝐺₂ the corresponding generators of 𝑁₃ are the following with 𝐹_𝛼,𝛽∶𝑁𝐺₁×𝑁𝐺₂→𝑁𝐺₃ and for 𝑛=3, 𝑝(𝑥)=𝑝₂𝑝₁𝑝₀(𝑥):
(i) 𝐹_(1,0)(2)𝑥₁,𝑦₂=𝑝𝑠₁𝑠₀𝑥₁,𝑠₂𝑦₂=𝑝₂𝑝₁𝑝₀𝑠₁𝑠₀𝑥₁,𝑠₂𝑦₂=𝑠₁𝑠₀𝑥₁−𝑠₂𝑠₁𝑥₁,𝑠₂𝑦₂, (11)
(ii) 𝐹_(2,0)(1)𝑥₁,𝑦₂=𝑝𝑠₂𝑠₀𝑥₁,𝑠₁𝑦₂=𝑝₂𝑝₁𝑝₀𝑠₂𝑠₀𝑥₁,𝑠₁𝑦₂=𝑠₂𝑠₀𝑥₁−𝑠₂𝑠₁𝑥₁,𝑠₁𝑦₂−𝑠₂𝑦₂. (12) For all 𝑥₂∈𝑁𝐺₂ and 𝑦₁∈𝑁𝐺₁ and considering the map 𝐹_𝛼,𝛽∶𝑁𝐺₂×𝑁𝐺₁→𝑁𝐺₃, the corresponding generator of 𝑁₃ is
(iii) 𝐹_(0)(2,1)𝑥₂,𝑦₁=𝑝𝑠₀𝑥₂,𝑠₂𝑠₁𝑦₁=𝑝₂𝑝₁𝑝₀𝑠₀𝑥₂,𝑠₂𝑠₁𝑦₁=𝑠₀𝑥₂−𝑠₁𝑥₂+𝑠₂𝑥₂,𝑠₂𝑠₁𝑦₁. (13) For all 𝑥₂, 𝑦₂∈𝑁𝐺₂ and for 𝐹_𝛼,𝛽∶𝑁𝐺₂×𝑁𝐺₂→𝑁𝐺₃ the corresponding generators of 𝑁₃ are
(iv) 𝐹₍₁₎₍₂₎𝑥₂,𝑦₂=𝑝𝑠₁𝑥₂,𝑠₂𝑦₂=𝑝₂𝑝₁𝑝₀𝑠₁𝑥₂,𝑠₂𝑦₂=𝑠₁𝑥₂−𝑠₂𝑥₂,𝑠₂𝑦₂, (14) (v) 𝐹₍₀₎₍₂₎𝑥₂,𝑦₂=𝑝𝑠₀𝑥₂,𝑠₂𝑦₂=𝑝₂𝑝₁𝑝₀𝑠₀𝑥₂,𝑠₂𝑦₂=𝑠₀𝑥₂,𝑠₂𝑦₂, (15)
(vı) 𝐹₍₀₎₍₁₎𝑥₂,𝑦₂=𝑝𝑠₀𝑥₂,𝑠₁𝑦₂=𝑝₂𝑝₁𝑝₀𝑠₀𝑥₂,𝑠₁𝑦₂=𝑠₀𝑥₂−𝑠₁𝑥₂,𝑠₁𝑦₂+𝑠₂𝑥₂,𝑠₂𝑦₂. (16)

Theorem 6. Let 𝑁𝐺₂ be a 2-dimensional Moore complex of a simplicial group 𝐆. Then 𝜕₂(𝑁𝐺₂)=[Ker𝑑₀,Ker𝑑₁] where 𝜕₂ is induced from 𝑑₂ by restriction.

Proof. For 𝑛=2, assume that 𝛼=(0), 𝛽=(1), and 𝑥₁, 𝑦₁∈𝑁𝐺₁=Ker𝑑₀. Now calculate 𝑑_𝑛𝐹_𝛼,𝛽.
Since 𝐹_(0),(1)(𝑥₁,𝑦₁)=[𝑠₀𝑥₁−𝑠₁𝑥₁,𝑠₁𝑦₁], from Proposition 5𝑑₂𝐹_(0),(1)𝑥₁,𝑦₁=𝑑₂𝑠₀𝑥₁−𝑑₂𝑠_1𝑖𝑑𝑥₁,𝑑₂𝑠_1𝑖𝑑𝑦₁=𝑠₀𝑑₁𝑥₁−𝑥₁,𝑦₁. (17)
At first we investigate whether 𝑠₀𝑑₁𝑥₁−𝑥₁ is in Ker𝑑₀ or not. 𝑑₀𝑠_0𝑖𝑑𝑑₁𝑥₁−𝑑₀𝑥₁₌₀=𝑑₁𝑥₁; (18) therefore 𝑠₀𝑑₁𝑥₁−𝑥₁∉Ker𝑑₀.
Secondly we examine whether 𝑠₀𝑑₁𝑥₁−𝑥₁ is in Ker𝑑₁ or not.
Since 𝑑₁𝑠_0𝑖𝑑𝑑₁𝑥₁−𝑑₁𝑥₁₌₀=𝑑₁𝑥₁−𝑑₁𝑥₁=0,𝑠₀𝑑₁𝑥₁−𝑥₁∈Ker𝑑₁. (19)
From the assumption 𝑦₁∈Ker𝑑₀, we get 𝐹_(0),(1)𝑥₁,𝑦₁∈Ker𝑑₁,Ker𝑑₀. (20)

Theorem 7. Let 𝑁𝐺₃ be a 3-dimensional Moore complex of a simplicial group 𝐆 with 26-adjacency. Then 𝜕₃𝑁𝐺₃⊆Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₁,Ker𝑑₀∩Ker𝑑₂+Ker𝑑₁∩Ker𝑑₂,Ker𝑑₀+Ker𝑑₁,Ker𝑑₀+Ker𝑑₀∩Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁, (21) where 𝜕₃ is induced from 𝑑₃ by restriction.

Proof. For 𝑛=3 investigate 𝑑_𝑛𝐹_𝛼,𝛽 where 𝑥₁∈𝑁𝐺₁ and 𝑦₂∈𝑁𝐺₂=Ker𝑑₀∩Ker𝑑₁.
From Proposition 5 we have 𝐹_(1,0)(2)(𝑥₁,𝑦₂)=[𝑠₁𝑠₀𝑥₁−𝑠₂𝑠₀𝑥₁,𝑠₂𝑦₂]. Then applying 𝑑₃ to 𝐹_(1,0)(2)(𝑥₁,𝑦₂), we get the following: 𝑑₃𝐹_(1,0)(2)𝑥₁,𝑦₂=𝑑₃𝑠₁𝑠₀𝑥₁−𝑑₃𝑠_2𝑖𝑑𝑠₀𝑥₁,𝑑₃𝑠_2𝑖𝑑𝑦₂=𝑠₁𝑑₂𝑠₀𝑥₁−𝑠₀𝑥₁,𝑦₂=𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁,𝑦₂. (22) Firstly, examine whether 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁ is in Ker𝑑₀ or not: 𝑑₀𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁=𝑑₀𝑠₁𝑠₀𝑑₁𝑥₁−𝑑₀𝑠₀𝑥₁=𝑠₀𝑑₀𝑠_0𝑖𝑑𝑑₁𝑥₁−𝑑₀𝑠_0𝑖𝑑𝑥₁=𝑠₀𝑑₁𝑥₁−𝑥₁. (23) So 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁∉Ker𝑑₀.
Secondly we investigate whether 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁ is in Ker𝑑₁ or not: 𝑑₁𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁=𝑑₁𝑠₁𝑠₀𝑑₁𝑥₁−𝑑₁𝑠₀𝑥₁=𝑑₁𝑠_1𝑖𝑑𝑠₀𝑑₁𝑥₁−𝑑₁𝑠_0𝑖𝑑𝑥₁=𝑠₀𝑑₁𝑥₁−𝑥₁. (24) Therefore 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁∉Ker𝑑₁.
Finally we check whether 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁ is in Ker𝑑₂ or not.
Since 𝑑₂𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁=𝑑₂𝑠_1𝑖𝑑𝑠₀𝑑₁𝑥₁−𝑑₂𝑠₀𝑥₁=𝑠₀𝑑₁𝑥₁−𝑠₀𝑑₁𝑥₁=0, (25) therefore 𝑠₁𝑠₀𝑑₁𝑥₁−𝑠₀𝑥₁∈Ker𝑑₂.
We get 𝐹_(0,1),(2)𝑥₁,𝑦₁∈Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁, (26) since 𝑦₂∈Ker𝑑₀∩Ker𝑑₁.
If 𝐹_(2,0)(1)𝑥₁,𝑦₂=𝑠₂𝑠₀𝑥₁−𝑠₂𝑠₁𝑥₁,𝑠₁𝑦₂−𝑠₂𝑦₂, (27) then 𝑑₃𝐹_(2,0)(1)𝑥₁,𝑦₂=𝑑₃𝑠₂𝑠₀𝑥₁−𝑠₂𝑠₁𝑥₁,𝑠₁𝑦₂−𝑠₂𝑦₂=𝑑₃𝑠_2𝑖𝑑𝑠₀𝑥₁−𝑑₃𝑠_2𝑖𝑑𝑠₁𝑥₁,𝑑₃𝑠₁𝑦₂−𝑑₃𝑠_2𝑖𝑑𝑦₂=𝑠₀𝑥₁−𝑠₁𝑥₁,𝑠₁𝑑₂𝑦₂−𝑦₂. (28)
At first we check whether 𝑠₀𝑥₁−𝑠₁𝑥₁ is in Ker𝑑₀, Ker𝑑₁, and Ker𝑑₂ or not. 𝑑₀𝑠₀𝑥₁−𝑠₁𝑥₁=𝑑₀𝑠_0𝑖𝑑𝑥₁−𝑑₀𝑠₁𝑥₁=𝑥₁−𝑑₀𝑠₁𝑥₁. (29) Thus 𝑠₀𝑥₁−𝑠₁𝑥₁∉Ker𝑑₀.
Next, since 𝑑₁𝑠₀𝑥₁−𝑠₁𝑥₁=𝑑₁𝑠_0𝑖𝑑𝑥₁−𝑑₁𝑠_1𝑖𝑑𝑥₁=𝑥₁−𝑥₁𝑠=0,₀𝑥₁−𝑠₁𝑥₁∈Ker𝑑₁, (30) and, finally, 𝑑₂𝑠₀𝑥₁−𝑠₁𝑥₁=𝑑₂𝑠₀𝑥₁−𝑑₂𝑠_1𝑖𝑑𝑥₁=𝑠₀𝑑₁𝑥₁−𝑥₁∉Ker𝑑₂. (31)
Now examine whether 𝑠₁𝑑₂𝑦₂−𝑦₂ is in Ker𝑑₀, Ker𝑑₁, and Ker𝑑₂ or not: 𝑑₀𝑠₁𝑑₂𝑦₂−𝑦₂=𝑑₀𝑠₁𝑑₂𝑦₂−𝑑₀𝑦₂=𝑠₀𝑑₀𝑑₂𝑦₂−𝑑₀𝑦₂=𝑠₀𝑑₁𝑑₀𝑦₂₌₀−𝑑₀𝑦₂₌₀=0. (32) Therefore 𝑠₁𝑑₂𝑦₂−𝑦₂∈Ker𝑑₀. We have the following: 𝑑₁𝑠₁𝑑₂𝑦₂−𝑦₂=𝑑₁𝑠_1𝑖𝑑𝑑₂𝑦₂−𝑑₁𝑦₂₌₀=𝑑₂𝑦₂∉Ker𝑑₁;𝑑₂𝑠₁𝑑₂𝑦₂−𝑦₂=𝑑₂𝑠_1𝑖𝑑𝑑₂𝑦₂−𝑑₂𝑦₂=𝑑₂𝑦₂−𝑑₂𝑦₂=0⟹𝑠₁𝑑₂𝑦₂−𝑦₂∈Ker𝑑₂. (33) So 𝐹_(2,0)(1)(𝑥₁,𝑦₂)=[Ker𝑑₁,Ker𝑑₀∩Ker𝑑₂].
For all 𝑥₂∈𝑁𝐺₂ and 𝑦₁∈𝑁𝐺₁ if 𝐹_(0)(2,1)𝑥₂,𝑦₁=𝑠₀𝑥₂−𝑠₁𝑥₂+𝑠₂𝑥₂,𝑠₁𝑠₁𝑦₁, (34) then 𝑑₃𝐹_(0)(2,1)𝑥₂,𝑦₁=𝑑₃𝑠₀𝑥₂−𝑠₁𝑥₂+𝑠₂𝑥₂,𝑠₂𝑠₁𝑦₁=𝑑₃𝑠₀𝑥₂−𝑑₃𝑠₁𝑥₂+𝑑₃𝑠_2𝑖𝑑𝑥₂,𝑑₃𝑠_2𝑖𝑑𝑠₁𝑦₁=𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂,𝑠₁𝑦₁. (35) Firstly investigate whether 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂ is in Ker𝑑₀, Ker𝑑₁, and Ker𝑑₂ or not: 𝑑₀𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂=𝑑₀𝑠_0𝑖𝑑𝑑₂𝑥₂−𝑑₀𝑠₁𝑑₂𝑥₂+𝑑₀𝑥₂₌₀=𝑑₂𝑥₂−𝑠₀𝑑₁𝑑₀𝑥₂₌₀=𝑑₂𝑥₂. (36) Thereby 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂∉Ker𝑑₀. We have 𝑑₁𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂=𝑑₁𝑠_0𝑖𝑑𝑑₂𝑥₂−𝑑₁𝑠_1𝑖𝑑𝑑₂𝑥₂+𝑑₁𝑥₂₌₀=𝑑₂𝑥₂−𝑑₂𝑥₂=0. (37) For this reason 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂∈Ker𝑑₁. We also have 𝑑₂𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂=𝑑₂𝑠₀𝑑₂𝑥₂−𝑑₂𝑠_1𝑖𝑑𝑑₂𝑥₂+𝑑₂𝑥₂=𝑠₀𝑑₁𝑑₂𝑥₂−𝑑₂𝑥₂+𝑑₂𝑥₂=𝑠₀𝑑₁𝑑₁𝑥₂₌₀=0. (38) Hence 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂∈Ker𝑑₂.
Later on we research whether 𝑠₂𝑠₁𝑦₁ is in Ker𝑑₀, Ker𝑑₁, and Ker𝑑₂ or not.
Since 𝑑₀(𝑠₁𝑦₁)=𝑠₀𝑑₀𝑦₁₌₀=0, 𝑠₁𝑦₁∈Ker𝑑₀.
Since 𝑑₁(𝑠₁𝑦₁)=𝑑₁𝑠_1𝑖𝑑𝑦₁=𝑦₁, 𝑠₁𝑦₁∉Ker𝑑₁.
Since 𝑑₂(𝑠₁𝑦₁)=𝑑₂𝑠_1𝑖𝑑𝑦₁=𝑦₁, 𝑠₁𝑦₁∉Ker𝑑₂.
Thus 𝑑₃𝐹_(0)(2,1)(𝑥₂,𝑦₁)∈[Ker𝑑₁∩Ker𝑑₂,Ker𝑑₀].
For all 𝑥₂, 𝑦₂∈𝑁𝐺₂ since 𝐹₍₀₎₍₂₎(𝑥₂,𝑦₂)=[𝑠₀𝑥₂,𝑠₂𝑦₂], 𝑑₃𝐹₍₀₎₍₂₎𝑥₂,𝑦₂=𝑑₃𝑠₀𝑥₂,𝑠₂𝑦₂=𝑑₃𝑠₀𝑥₂,𝑑₃𝑠_2𝑖𝑑𝑦₂=𝑠₀𝑑₂𝑥₂,𝑦₂. (39) By using properties of the commutator we have 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂,𝑦₂=𝑠₀𝑑₂𝑥₂+𝑥₂−𝑠₁𝑑₂𝑥₂,𝑦₂=𝑠₀𝑑₂𝑥₂,𝑦₂+𝑥₂−𝑠₁𝑑₂𝑥₂,𝑦₂,𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂,𝑦₂+𝑦₂,𝑥₂−𝑠₁𝑑₂𝑥₂=𝑠₀𝑑₂𝑥₂,𝑦₂=𝑑₃𝐹₍₀₎₍₂₎𝑥₂,𝑦₂. (40) Thus 𝑑₃𝐹₍₀₎₍₂₎𝑥₂,𝑦₂∈Ker𝑑₁∩Ker𝑑₂,Ker𝑑₁∩Ker𝑑₀+Ker𝑑₀∩Ker𝑑₁,Ker𝑑₀∩Ker𝑑₂. (41) If 𝐹₍₁₎₍₂₎(𝑥₂,𝑦₂)=[𝑠₁𝑥₂−𝑠₂𝑥₂,𝑠₂𝑦₂], then 𝑑₃𝐹₍₁₎₍₂₎𝑥₂,𝑦₂=𝑑₃𝑠₁𝑥₂−𝑠₂𝑥₂,𝑠₂𝑦₂=𝑑₃𝑠₁𝑥₂−𝑑₃𝑠_2𝑖𝑑𝑥₂,𝑑₃𝑠_2𝑖𝑑𝑦₂=𝑠₁𝑑₂𝑥₂−𝑥₂,𝑦₂. (42) Firstly we check whether 𝑠₁𝑑₂𝑥₂−𝑥₂ is in Ker𝑑₀, Ker𝑑₁, and Ker𝑑₂ or not: 𝑑₀𝑠₁𝑑₂𝑥₂−𝑥₂=𝑑₀𝑠₁𝑑₂𝑥₂−𝑑₀𝑥₂₌₀=𝑠₀𝑑₀𝑑₂𝑥₂=𝑠₀𝑑₁𝑑₀𝑥₂₌₀=0. (43) Therefore 𝑠₁𝑑₂𝑥₂−𝑥₂∈Ker𝑑₀. Since 𝑑₁𝑠₁𝑑₂𝑥₂−𝑥₂=𝑑₁𝑠_1𝑖𝑑𝑑₂𝑥₂−𝑑₁𝑥₂₌₀=𝑑₂𝑥₂, (44)𝑠₁𝑑₂𝑥₂−𝑥₂∉Ker𝑑₁. We have 𝑑₂𝑠₁𝑑₂𝑥₂−𝑥₂=𝑑₂𝑠_1𝑖𝑑𝑑₂𝑥₂−𝑑₂𝑥₂=𝑑₂𝑥₂−𝑑₂𝑥₂=0. (45) Hence 𝑠₁𝑑₂𝑥₂−𝑥₂∈Ker𝑑₂.
Because of the case 𝑦₂∈Ker𝑑₀∩Ker𝑑₁, 𝑑₃𝐹₍₁₎₍₂₎𝑥₂,𝑦₂∈Ker𝑑₀∩Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁. (46) If 𝐹₍₀₎₍₁₎(𝑥₂,𝑦₂)=[𝑠₀𝑥₂−𝑠₁𝑥₂,𝑠₁𝑦₂]+[𝑠₂𝑥₂,𝑠₂𝑦₂], then 𝑑₃𝐹₍₀₎₍₁₎𝑥₂,𝑦₂=𝑑₃𝑠₀𝑥₂−𝑑₃𝑠₁𝑥₂,𝑠₁𝑦₂+𝑑₃𝑠_2𝑖𝑑𝑥₂,𝑑₃𝑠_2𝑖𝑑𝑦₂=𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂,𝑠₁𝑑₂𝑦₂+𝑥₂,𝑦₂. (47) Consider the following commutator: 𝑠₀𝑑₂𝑥₂−𝑠₁𝑑₂𝑥₂+𝑥₂,𝑠₁𝑑₂𝑦₂−𝑦₂, (48) and code the terms of this commutator such as 𝑎=𝑠₀𝑑₂𝑥₂,𝑏=𝑠₁𝑑₂𝑦₂,𝑐=𝑠₁𝑑₂𝑥₂,𝑑=𝑥₂,𝑒=𝑦₂, (49) in order to simplify the algebraic operations. Thus, by using the properties and definition of the commutator we obtain the following: []=[]+[],()−()−{}.𝑎−𝑐+𝑑,𝑏−𝑒𝑎−𝑐,𝑏𝑑,𝑒𝑎−𝑐+𝑑)(𝑏−𝑒𝑏−𝑒)(𝑎−𝑐+𝑑=𝑎𝑏−𝑐𝑏+𝑑𝑏−𝑎𝑒+𝑐𝑒−𝑑𝑒𝑏𝑎−𝑏𝑐+𝑏𝑑−𝑒𝑎+𝑒𝑐−𝑒𝑑 (50) Consider the following cases: ()()=[],()()=[].𝑎𝑏−𝑐𝑏−𝑏𝑎+𝑏𝑐=𝑎−𝑐𝑏−𝑏𝑎−𝑐𝑎−𝑐,𝑏𝑐𝑒−𝑑𝑒−𝑒𝑐+𝑒𝑑=𝑐−𝑑𝑒−𝑒𝑐−𝑑𝑐−𝑑,𝑒 (51) And from the remaining terms we get []()−()𝑑=[].𝑑𝑏−𝑏𝑑−𝑑,𝑒=𝑑𝑏−𝑏𝑑−𝑑𝑒+𝑒𝑑=𝑑𝑏−𝑒𝑏−𝑒𝑑,𝑏−𝑒 (52) Consequently for 𝑛=3 we have 𝜕₃𝑁𝐺₃⊆Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₁,Ker𝑑₀∩Ker𝑑₂+Ker𝑑₁∩Ker𝑑₂,Ker𝑑₀+Ker𝑑₁,Ker𝑑₀+Ker𝑑₀∩Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁. (53)

Corollary 8. Let 𝑁𝐺₃ be a 3-dimensional Moore complex of a simplicial group 𝐆 with 26-adjacency. Then 𝜕₃𝑁𝐺₃⊆Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₁,𝐾𝑒𝑟𝑑₀∩Ker𝑑₂+Ker𝑑₁∩Ker𝑑₂,Ker𝑑₀+Ker𝑑₁,Ker𝑑₀+Ker𝑑₀∩Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁. (54)

Proof. Otherwise inclusion for the previous theorem is obtained from [4, 5]. Therefore 𝜕₃𝑁𝐺₃=Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₁,Ker𝑑₀∩Ker𝑑₂+Ker𝑑₁∩Ker𝑑₂,Ker𝑑₀+Ker𝑑₁,Ker𝑑₀+Ker𝑑₀∩Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁+Ker𝑑₂,Ker𝑑₀∩Ker𝑑₁. (55)

3. Conclusion

In this paper for dimension 2 and dimension 3, we obtained the Moore complex of simplicial groups generated by hypercrossed complex pairings in digital images.