Lectures on Geometry (original) (raw)
Increasing Functions and Closure Operations on Generalized Ordered Sets
2014
In this paper, having in mind Galois connections, we establish several consequences of the following definitions. An ordered pair X (≤) = ( X , ≤ ) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set). A function f of one goset X to another Y is called increasing if u ≤ v implies f(u) ≤ f(v) for all u, v ∈ X. In particular, an increasing function φ of X to itself is called a closure operation on X if x ≤ φ(x) and φ φ(x) ≤ φ(x) for all x ∈ X. The results obtained extend and supplement some former results on increasing functions and closure operations, and can be generalized to relator spaces.
An order-theoretic perspective on categorial closure operators
Quaestiones Mathematicae, 2018
This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco's allow one to see more easily the connections between standard conditions on general cco's, and furthermore, we show that these connections for cco's can be even deduced from the corresponding ones for bco's, when considering cco's relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco's via bco's is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco's, independent results on cco's are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco's also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.
Topogenous orders and closure operators on posets
Turkish journal of mathematics, 2024
We introduce the notion of topogenous orders on a poset X to be certain endomaps on X. We build on a Galois connection between endomaps and binary relations on X and study relationships between endomap properties and corresponding relational properties. In particular, we determine the topogenous orders that are in a one-to-one correspondence with (idempotent) closure operators. Extending our considerations to the categorical level, we find a cartesian closed category of topogenous systems.
Transformations of discrete closure systems
Acta Mathematica Hungarica, 2013
Discrete systems such as sets, monoids, groups are familiar categories. The internal structre of the latter two is defined by an algebraic operator. In this paper we describe the internal structure of the base set U by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggest the wide applicability of closure systems. Next, we look at functions f which map power sets 2 U into power sets 2 U ′ , which we call transformations. In particular, we consider transformations which are monotone, continuous, or closed. These can be used to establish criteria for asserting that "the closure of a transformed image under f is equal to the transformed image of the closure". Finally, we show that the categories MCont and MClo of closure systems with morphisms given by the monotone continuous transformations and monotone closed transformations respectively have concrete direct products. And the supercategory Clo of MClo whose morphisms are just the closed transformations is shown to be even cartesian closed. * The second author acknowledges a partial financial support from the Ministry of Education of the Czech Republic, project no. MSM 0021630518.
Galois-type connections and closure operations on preordered sets
For a function f of one preordered set X to another Y , we shall establish several consequences of the following two definitions: (a) f is increasingly ϕ-regular, for some function ϕ of X to itself, if for any x 1 , x 2 ∈ X we have x 1 ≤ ϕ (x 2) if and only if f (x 1) ≤ f (x 2); (b) f is increasingly g-normal, for some function g of Y to X , if for any x ∈ X and y ∈ Y we have f (x) ≤ y if and only if x ≤ g (y). These definitions have been mainly suggested to us by a recent theory of relators (families of relations) worked out byÁ. Száz and G. Pataki and the extensive literature on Galois connections and residuated mappings.
New Approach for Closure Spaces by Relations
2000
Recently, the general topology has become the appropriated frame for every collection related to relations because topology is required not only for mathematics and physics but also for biology, rough set theory, biochemistry, and dynamics. In this paper, we have introduced another concept of the closure operator. In so doing, the idempotent condition, which has never been realized, is achieved.
2012
This paper has as its main objective a critica review of typical classifications based on the disciplines of application (e.g.(e.g.(e.g. topology, algebra, geomerrv) and their particular needs. Special attention is given to the example of geometric closure spaces and to the question what property 0/cdot0/\cdot0/cdot properties should be used to distinguish t/list/list/lis category. JnJnJn this context, new class of closure spaces of character nnn is introduced. Arguments are provided that geometric closure spaces should be distinguished as closure spaces of character 2. Also, some characteristics of closure spaces of character are given. Finally, the exchange prope/typrope/typrope/ty of closure spaces which is usually considered as defining for geometric closure spaces is associated with the issue of diSbackslashintointdiS \backslash \int ointdiSbackslashintoint union decomposability of closure spaces. Some suggestions are made regarding more meaningful, comprehensliota’ecomprehensl\iota’ ecomprehensliota’e classification of closure downarrow\downarrowdownarrowspaces.
An Axiomatic Characterization of Linear Orders
The Review of Modern Logic, 2004
In this paper we propose an axiomatic characterization of the set of linear orders using the concept of a choice rule, which assigns to each ordered pair of feasible alternatives and a reflexive binary relation, exactly one element from the feasible set.