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Jordan Pairs of Adjacency Relations in Digital Spaces
Author(s) -
HERMAN GABOR T.
Publication year - 1994
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1994.tb44141.x
Subject(s) - adjacency list , partition (number theory) , surface (topology) , boundary (topology) , binary number , mathematics , complement (music) , binary relation , connected component , connected space , component (thermodynamics) , set (abstract data type) , space (punctuation) , combinatorics , relation (database) , computer science , pure mathematics , discrete mathematics , geometry , topological space , mathematical analysis , arithmetic , database , physics , chemistry , operating system , biochemistry , thermodynamics , programming language , complementation , gene , phenotype
We define a digital space to be a pair consisting of an arbitrary nonempty set V and a symmetric binary relation π on V with respect to which V is connected. Our intent is to capture and investigate the notion of a boundary in this extremely general environment. Our motivation comes from practical applications, where boundaries need to be identified in multidimensional data sets with the further aim of displaying them on a computer screen. Our definitions are biased towards such applications. In particular, we refer to elements of V as spels (short for “spatial elements”). We call an arbitrary nonempty subset of a π surface and we define the notions of the interior and the exterior of a surface. We also introduce the notion of a near‐Jordan surface; its interior and exterior partition V. We call a symmetric binary relations on V that contains π a spel‐adjacency . For spel‐adjacencies κ and λ, we call a surface κλ‐Jordan if it is near‐Jordan, its interior is κ‐connected and its exterior is λ‐connected. Our first main result is that the surface between a κ‐connected set and any λ‐component of its complement is κλ‐Jordan. We introduce the notion of a binary picture in which there is an assignment of a 1 or a 0 to elements of V. A κλ‐boundary is defined as the surface between a λ‐component of 1‐spels and a λ‐component of 0‐spels. We discuss the notion of tightness for spel‐adjacencies. Our next main result is that for tight spel‐adjacencies κ and λ, the interior of a κλ‐boundary is κ‐connected, while its exterior is λ‐connected. We say that {κ, λ} is a Jordan pair , if κ and λ are tight spel‐adjacencies and every κλ‐boundary (equivalently, every λκ‐boundary) is near‐Jordan. It therefore follows from the previously stated result that if {κ, λ} is a Jordan pair, then a κλ‐boundary is κλ‐Jordan. We also show that if {κ, λ} is a Jordan pair and κ′ and λ′ are tight spel adjacencies containing κ and λ, respectively, then {κ′, λ′} is also a Jordan pair.