Affiliations: Institute of Mathematics, Brno University of Technology, 616 69 Brno, Czech Republic. [email protected]
Correspondence:
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This work was supported by Brno University of Technology from the Specific Research Program, project no. FSI-S-20-6187. Address for correspondence: Institute of Mathematics, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic.
Abstract: In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.
Keywords: Digital space, n-ary relation, closure operator, connectedness, digital Jordan curve, digital Jordan surface
