Abstract
Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.
Highlights
We further investigate some properties of the Isok ( X )-action on ( X, k) and explore a certain condition that makes the alignment of fixed point sets by the Conk ( X )-action perfect, where for a digital object ( X, k) we say that F (Conk ( X )) is perfect if it is equal to [0, X ] ]Z
One of the important things that comes from this paper is the development of group structures, Isokn,l which play crucial roles in studying digital images from the viewpoint of fixed point theory
We have intensively studied the algebraic topological structures of Isokn,l (·) and
Summary
One important thing to note is that the current approach in a digital topological setting plays an important role in applied mathematics as well as applied sciences Based on this approach, let Dk ( X ) ∈ { Isok ( X ), Conk ( X )}, the present paper explores some new structures and features associated with the fixed point sets by the Dk ( X )-actions on a digital image ( X, k) (see Definition 10) and further, examines if alignments of fixed point sets by the Dk ( X )-actions are 2-connected or perfect (see Definition 11). We will use the notation X ] to indicate the cardinality of a set X
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