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We introduce a connectedness in the digital space ℤ3 induced by a quaternary relation. Using this connectedness, we prove a digital 3D Jordan-Brouwer separation theorem for boundary surfaces of the digital polyhedra that may be face-to-face tiled with certain digital tetrahedra in ℤ3. An advantage of the digital Jordan surfaces obtained over those given by the Khalimsky topology is that the former may bend at the acute dihedral angle π4{\pi \over 4}.

After introducing a graph connectedness induced by a given set of paths of the same length, we focus on the 2-adjacency graph on the digital line Z{\mathbb{Z}} with a certain set of paths of length nn for every positive integer nn. The connectedness in the strong product of three copies of the graph is used to define digital Jordan surfaces. These are obtained as polyhedral surfaces bounding the polyhedra that can be face-to-face tiled with digital tetrahedra.

We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α>0\alpha \gt 0 in such a way that the closure of a set AA is given by closures of certain α\alpha -indexed sequences formed by points of AA. It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.

Given a simple graph with the vertex set X, we discuss a closure operator on X induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line ℤ and consider the closure operators on ℤm (m a positive integer) that are induced by a special product of m copies of the introduced set of paths. We focus on the case m = 3 and show that the closure operator considered provides the digital space ℤ3 with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.

For every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

We introduce and study a new operation of product of universal hyperalgebras which lies, with respect to set inclusion, between the cartesian product of the hyperalgebras and the cartesian product of their idempotent hulls. We give sufficient conditions for the validity of the first exponential law and a weak form of the second exponential law for the direct power of universal hyperalgebras with respect to the product introduced.

We discuss certain ternary relations, called plain, and show that each of them induces a connectedness on its underlying set. This connectedness allows for definitions of concepts of simple closed and Jordan curves. We introduce a particular plain ternary relation on the digital plane ℤ2 and, as the main result, we prove a digital analogue of the Jordan curve theorem for the connectedness induced by this relation. It follows that the ternary relation introduced may be used as a convenient structure on the digital plane for the study of the geometric properties of digital images that are related to boundaries because boundaries of objects in digital images are represented by digital Jordan curves. An advantage of this structure over the Khalimsky topology is that it allows Jordan curves to turn at the acute angle π/4 at some points.

Departing from a categorical concept of topogenous orders defined relative to the bifibration of subobjects, we introduce and discuss topogenous orders on forms, i.e., faithful and amnestic functors. These topogenous orders are shown to include both closure and interior operators on forms. We define and study two special morphisms relative to a topogenous order, namely strict and final morphisms. We give a characterization of the two morphisms by the help of their cartesian and cocartesian liftings. Some examples from topology and algebra demonstrating our results are included. [ABSTRACT FROM AUTHOR]

We employ closure operators associated with n-ary relations, n > 1 an integer, to provide the digital space ℤ3 with connectedness structures. We show that each of the six inscribed tetrahedra obtained by canonical tessellation of a digital cube in ℤ3 with edges consisting of 2n − 1 points is connected. This result is used to prove that certain bounding surfaces of the polyhedra in ℤ3 that may be face-to-face tiled with such tetrahedra are digital Jordan surfaces (i.e., separate ℤ3 into exactly two connected components). An advantage of these Jordan surfaces over those with respect to the Khalimsky topology is that they may possess acute dihedral angles π 4 while, in the case of the Khalimsky topology, the dihedral angles may never be less than π 2 . [ABSTRACT FROM AUTHOR]

Several naturally defined lower separation axioms for bitopological spaces obtained by modifying the axioms T0, T1, and R0 appear in the literature. We introduce and study analogous separation axioms for bitopogenous spaces. In particular, we investigate relationships between the axioms and discuss the conditions under which the relationships are similar to those between the corresponding separation axioms for bitopological spaces. [ABSTRACT FROM AUTHOR]

We introduce a graph connectedness induced by a given set of paths of the same length. We focus on the 2-adjacency graph on the digital line ℤ with a certain set of paths of length n for every positive integer n. The connectedness in the strong product of two and three copies of the graph is used to define digital Jordan curves and digital Jordan surfaces, respectively. Such definitions build on an edge-to-edge tiling with triangles in the digital plane and a face-to-face tiling by cubes, prisms and pyramids in the (3D) digital space, respectively. [ABSTRACT FROM AUTHOR]