Topogenous orders and closure operators on posets (original) (raw)
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.
Some Results on Ordered Structures in Toposes
A b s t r a c t. A topos version of Cantor's back and forth theorem is established and used to prove that the ordered structure of the rational numbers Q, < is homogeneous in any topos with natural numbers object. The notion of effective homogeneity is introduced, and it is shown that Q, < is a minimal effectively homogeneous structure, that is, it can be embedded in every other effectively homogeneous ordered structure.
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
A Spectral-style Duality for Distributive Posets
Order, 2017
In this paper, we present a topological duality for a category of partially ordered sets that satisfy a distributivity condition studied by David and Erné. We call these posets mo-distributive. Our duality extends a duality given by David and Erné because our category of spaces has the same objects as theirs but the class of morphisms that we consider strictly includes their morphisms. As a consequence of our duality, the duality of David and Erné easily follows. Using the dual spaces of the mo-distributive posets we prove the existence of a particular 1-completion for mo-distributive posets that might be different from the canonical extension. This allows us to show that the canonical extension of a distributive meet-semilattice is a completely distributive algebraic lattice.