Topogenous orders and closure operators on posets (original) (raw)
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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.
Journal of Computer and System Sciences, 1980
characterization of some constructions relating to ordered and complete algebras is given. In particular the completion of a poset and the initial objects of semivarieties of (complete) ordered algebras are studied. The quotient construction allows a reconciliation of the equational and synthetic (order-theoretic) approaches to abstract data types.
Some examples of universal and generic partial orders
Contemporary Mathematics, 2011
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order for various combinatorial objects.
Working Document, 2022
We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice ---specifically that all partial equivalence relations have an index--- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice. We reformulate and generalise a number of theorems originally due to Riguet on polar coverings of a relation. We study the properties of the ``diagonal'' of a relation (called the ``diff\'{e}rence'' by Riguet who introduced the concept in 1951). In particular, we formulate and prove a general theorem relating properties of the diagonal of a relation to block-ordered relations; the theorem generalises a property that Riguet called an ``analogie frappante'' between the ``diff\'{e}rence'' of a relation and ``relations de Ferrers'' (a special case of block-ordered relations).
eb 2 00 0 Algebraic duality for partially ordered sets
2000
For an arbitrary partially ordered set P its dual P ∗ is built as the collection of all monotone mappings P → 2 where 2 = {0, 1} with 0 < 1. The set of mappings P ∗ is proved to be a complete lattice with respect to the pointwise partial order. The second dual P ∗∗ is built as the collection of all morphisms of complete lattices P ∗ → 2 preserving universal bounds. Then it is proved that the partially ordered sets P and P ∗∗ are isomorphic. AMS classification: 06A06, 06A15 Introduction The results presented in this paper can be considered as the algebraic counterpart of the duality in the theory of linear spaces. The outline of the construction looks as follows. Several categories occur in the theory of partially ordered sets. The most general is the category POSET whose objects are partially ordered sets and the morphisms are the monotone mappings. Another category which will be used is BCL whose objects are (bounded) complete lattices and the morhisms are the lattice homomorphi...
Tychonoff Poset Structures and Auxiliary Relations
Annals of the New York Academy of Sciences, 1995
where <,a are the understood partial order and auxiliary relation on our space. DEFINITION 1.1: (1) X U y =+ x s y , (4 is no stronger than s), (2) w s x u y a z + W Q z, (a is compatible with s), (3) X U y-(3 z) (x < z 4 y) (4 has the interpolation property). An auxiliary relation 4 on X is: approximating if for each x E X , v(u x) = x , and subdirecting if for each x E X , 4 x is directed by a. An auxiliary relation on a partially ordered set (X, s) is a relation a on X satisfying: FACTS ABOUT AUXILIARY RELATIONS: (a) Our definitions differ somewhat from those in the literature; in particular, [3] and others require that a b e subdirecting, but d o not require the interpolation property. However, the interpolation property holds for all auxiliary relations used in practice. W e avoid the assumption that auxiliary relations are subdirecting to exploit in the remainder of this paper the connection between auxiliary relations and quasiproximities noted in [17]. This simplifies represention of the topologies arising from X with i and 4. These are: u4, the topology generated by {$ x I x E X), and o, the topology generated by {X\T x I x E x). (b) By (1) and (2), 4 is transitive, for i f x U yU z then by (1) and the reflexivity In many key cases mentioned below, 4 is neither reflexive nor irreflexive. (c) Notice that the definition of approximating auxiliary relation doesn't actually require the existence of suprema, since v(# x) = x o (v y 2 x) (3 ZU X) (Z $ y). There are three rather standard exam-of s, x s x 4 y i z, S O by (2), x u Z. EXAMPLES OF AUXILIARY RELATIONS: ples of auxiliary relations which w e follow throughout the paper: EXAMPLE (a): Certainly, the simplest example is 4 = s. It is approximating, and is trivially subdirecting. It is also certainly the largest possible auxiliary relation on a poset. us is the topology generated by all the ? x ; thus the collection of all sets closed under s. It is called the Alexandroff topology resulting from s. In particular, for the partial order = , u= is the discrete topology (and o is the cofinite topology). EXAMPLE (b): T h e way-below relation, <<, which plays a central role in the theory of continuous lattices is defined and thoroughly discussed in [3]; a more
Factorisations of some partially ordered sets and small categories
2021
Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that preserves the antisymmetry of the order Relation. Finally some suggestions are given, how the orbit categories can be represented by simple directed and annotated graphs and annotated binary relations. These relations are reflexive, and, in many cases, they can be chosen to be antisymmetric. From these constructions arise different suggestions for fundamental systems of partially ordered sets and reconstruction data which are illustrated by examples from mathematical music theory.
On Colimits Over Arbitrary Posets
Glasgow Mathematical Journal, 2015
We characterize those partially ordered sets I for which the canonical maps M i → colim M j into colimits of abstract sets are always injective, provided that the transition maps are injective. We also obtain some consequences for colimits of vector spaces. 2010 Mathematics Subject Classification. 18A30, 06A06. 1. Introduction. Crowns arise in various problems related to partially ordered sets (posets). Thus, for example, they appear in the study of retracts and fixed points (see [7]), in calculation of the cohomological dimension (see [5]), in applications to homotopy theory (see [11]) and in the investigation of incidence algebras and their quotients (see [1] and [6]). At the same time, quite often they play a "negative" role: the absence of crowns of some kind ensures the existence of certain good properties of posets or constructions related to them. For instance, an incidence algebra κ[S] of a finite poset S is completely separating if and only if S contains no crowns [6]. It is not surprising that such a situation arises in a problem of colimits which is discussed in this note: roughly speaking, the crowns are antagonists of directed sets for which colimits are usually considered and well understood (note that colimits over directed posets are called directed colimits, or direct limits, or inductive limits). More precisely, if one takes a directed colimit colim M i (also denoted by lim − → M i), where i runs over a directed poset I, such that the transition maps ϕ ij : M i → M j are injective, then the canonical maps M i → colim M j are also injective, which is a crucial property for applications. Thus, one may wonder which are the posets I for which this always happens. We completely characterize such I in the case when the M i 's are abstract sets and obtain consequences for the colimits of vector spaces. This problem is related to a similar question about the ring-theoretic version of cross-sectional algebras of Fell bundles over inverse semigroups studied in [8], since such algebras are epimorphic images of colimits of vector spaces over non-necessarily directed posets. In the C *-algebraic context, it is proved that the fibres are canonically embedded into the cross-sectional algebra; however, the abstract ring theoretic version
Ordered Algebraic Structures and Related Topics
Contemporary Mathematics, 2017
We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.
Semigroup Forum, 2005
In this paper, we consider some category-theoretic properties of the category Pos -S of all S -posets (posets equipped with a compatible right action of a pomonoid S ), with monotone action-preserving maps between them. We first discuss some general category-theoretic ingredients of Pos -S ; specifically, we characterize several kinds of epimorphisms and monomorphisms. Then, we present some adjoint relations of Pos -S with Pos , Set , and Act -S . In particular, we discuss free and cofree objects. We also examine other categorytheoretic properties, such as cartesian closedness and monadicity. Finally, we consider projectivity in Pos -S with respect to regular epimorphisms and show that it is the same as projectivity, although projectives are not generally retracts of free objects over posets.