Better-Quasi-Order: Ideals and Spaces
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A “Three-Sentence Proof” of Hansson's Theorem
Econ Theory Bull (2018) 6:111–114 https://doi.org/10.1007/s40505-017-0127-2 RESEARCH ARTICLE A “three-sentence proof” of Hansson’s theorem Henrik Petri1 Received: 18 July 2017 / Accepted: 28 August 2017 / Published online: 5 September 2017 © The Author(s) 2017. This article is an open access publication Abstract We provide a new proof of Hansson’s theorem: every preorder has a com- plete preorder extending it. The proof boils down to showing that the lexicographic order extends the Pareto order. Keywords Ordering extension theorem · Lexicographic order · Pareto order · Preferences JEL Classification C65 · D01 1 Introduction Two extensively studied binary relations in economics are the Pareto order and the lexicographic order. It is a well-known fact that the latter relation is an ordering exten- sion of the former. For instance, in Petri and Voorneveld (2016), an essential ingredient is Lemma 3.1, which roughly speaking requires the order under consideration to be an extension of the Pareto order. The main message of this short note is that some fundamental order extension theorems can be reduced to this basic fact. An advantage of the approach is that it seems less abstract than conventional proofs and hence may offer a pedagogical advantage in terms of exposition. Mandler (2015) gives an elegant proof of Spzilrajn’s theorem (1930) that stresses the importance of the lexicographic I thank two anonymous referees for helpful comments. Financial support by the Wallander–Hedelius Foundation under Grant P2014-0189:1 is gratefully acknowledged. B Henrik Petri [email protected] 1 Department of Finance, Stockholm School of Economics, Box 6501, 113 83 Stockholm, Sweden 123 112 H. -
Strategies for Linear Rewriting Systems: Link with Parallel Rewriting And
Strategies for linear rewriting systems: link with parallel rewriting and involutive divisions Cyrille Chenavier∗ Maxime Lucas† Abstract We study rewriting systems whose underlying set of terms is equipped with a vector space structure over a given field. We introduce parallel rewriting relations, which are rewriting relations compatible with the vector space structure, as well as rewriting strategies, which consist in choosing one rewriting step for each reducible basis element of the vector space. Using these notions, we introduce the S-confluence property and show that it implies confluence. We deduce a proof of the diamond lemma, based on strategies. We illustrate our general framework with rewriting systems over rational Weyl algebras, that are vector spaces over a field of rational functions. In particular, we show that involutive divisions induce rewriting strategies over rational Weyl algebras, and using the S-confluence property, we show that involutive sets induce confluent rewriting systems over rational Weyl algebras. Keywords: confluence, parallel rewriting, rewriting strategies, involutive divisions. M.S.C 2010 - Primary: 13N10, 68Q42. Secondary: 12H05, 35A25. Contents 1 Introduction 1 2 Rewriting strategies over vector spaces 4 2.1 Confluence relative to a strategy . .......... 4 2.2 Strategies for traditional rewriting relations . .................. 7 3 Rewriting strategies over rational Weyl algebras 10 3.1 Rewriting systems over rational Weyl algebras . ............... 10 3.2 Involutive divisions and strategies . .............. 12 arXiv:2005.05764v2 [cs.LO] 7 Jul 2020 4 Conclusion and perspectives 15 1 Introduction Rewriting systems are computational models given by a set of syntactic expressions and transformation rules used to simplify expressions into equivalent ones. -
Bijective Link Between Chapoton's New Intervals and Bipartite Planar Maps
Bijective link between Chapoton’s new intervals and bipartite planar maps Wenjie Fang* LIGM, Univ. Gustave Eiffel, CNRS, ESIEE Paris F-77454 Marne-la-Vallée, France June 18, 2021 Abstract In 2006, Chapoton defined a class of Tamari intervals called “new intervals” in his enumeration of Tamari intervals, and he found that these new intervals are equi- enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called “degree tree”. Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy. 1 Introduction On classical Catalan objects, such as Dyck paths and binary trees, we can define the famous Tamari lattice, first proposed by Dov Tamari [Tam62]. This partial order was later found woven into the fabric of other more sophisticated objects. A no- table example is diagonal coinvariant spaces [BPR12, BCP], which have led to several generalizations of the Tamari lattice [BPR12, PRV17], and also incited the interest in intervals in such Tamari-like lattices. Recently, there is a surge of interest in the enu- arXiv:2001.04723v2 [math.CO] 16 Jun 2021 meration [Cha06, BMFPR11, CP15, FPR17] and the structure [BB09, Fan17, Cha18] of different families of Tamari-like intervals. In particular, several bijective rela- tions were found between various families of Tamari-like intervals and planar maps [BB09, FPR17, Fan18]. The current work is a natural extension of this line of research. In [Cha06], other than counting Tamari intervals, Chapoton also introduced a subclass of Tamari intervals called new intervals, which are irreducible elements in a grafting construction of intervals. -
A New Order Theory of Set Systems and Better Quasi-Orderings
03 Progress in Informatics, No. 9, pp.9–18, (2012) 9 Special issue: Theoretical computer science and discrete mathematics Research Paper A new order theory of set systems and better quasi-orderings Yohji AKAMA1 1Mathematical Institute, Tohoku University ABSTRACT By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory of this new order type and continuous, monotone function between set systems corresponds to the theory of well quasi-orderings (WQOs). As Nash-Williams developed the theory of WQOs to the theory of better quasi- orderings (BQOs), we introduce a set system that has order type and corresponds to a BQO. We prove that the class of set systems corresponding to BQOs is closed by any monotone function. In (Shinohara and Arimura. “Inductive inference of unbounded unions of pattern languages from positive data.” Theoretical Computer Science, pp. 191–209, 2000), for any set system L, they considered the class of arbitrary (finite) unions of members of L.Fromview- point of WQOs and BQOs, we characterize the set systems L such that the class of arbitrary (finite) unions of members of L has order type. The characterization shows that the order structure of the set system L with respect to the set inclusion is not important for the result- ing set system having order type. We point out continuous, monotone function of set systems is similar to positive reduction to Jockusch-Owings’ weakly semirecursive sets. -
Normal Tree Orders for Infinite Graphs
Normal tree orders for infinite graphs J.-M. Brochet and R. Diestel A well-founded tree T defined on the vertex set of a graph G is called normal if the endvertices of any edge of G are comparable in T . We study how normal trees can be used to describe the structure of infinite graphs. In particular, we extend Jung’s classical existence theorem for trees of height ω to trees of arbitrary height. Applications include a structure theorem for graphs without large complete topological minors. A number of open problems are suggested. 1. Introduction: normal spanning trees The aim of this paper is to see how a classical and powerful structural device for the study of countable graphs, the notion of a normal spanning tree, can be made available more generally. The existence of such spanning trees, while trivial in the finite case (where they are better known as depth-first search trees), is in general limited to countable graphs. By generalizing the graph theoretical trees involved to order theoretical trees, a concept better suited to express uncountably long ‘ends’, we shall be able to extend the classical existence theorems for normal trees to arbitrary cardinalities, while retaining much of their original strength. Throughout the paper, G will denote an arbitrary connected graph. Con- sider a tree T ⊆ G, with a root r, say. If T spans G, the choice of r imposes a partial order on the vertex set V (G) of G: write x 6 y if x lies on the unique r–y path in T . -
BETTER QUASI-ORDERS for UNCOUNTABLE CARDINALS T
Sh:110 ISRAEL JOURNAL OF MATHEMATICS, Vol. 42, No. 3, 1982 BETTER QUASI-ORDERS FOR UNCOUNTABLE CARDINALS t BY SAHARON SHELAH ABSTRACT We generalize the theory of Nash-Williams on well quasi-orders and better quasi-orders and later results to uncountable cardinals. We find that the first cardinal K for which some natural quasi-orders are K-well-ordered, is a (specific) mild large cardinal. Such quasi-orders are ~ (the class of orders which are the union of <- A scattered orders) ordered by embeddability and the (graph theoretic) trees under embeddings taking edges to edges (rather than to passes). CONTENTS §0. Introduction ................................................................................................................. 178 ]Notation, including basic definitions and claims on barriers.] §1. The basic generalization ............................................................................................... 186 [We define K-X-bqo, and prove it is preserved by ~a (Q).] §2. Existence theorem and a stronger notion suitable for powers ....................................... 193 [We define the relevant large cardinals, and prove the existence of non-trivially K-X-bqos, prove that the narrowness cardinal of a Q is some large cardinal and introduce e.g. "[K]-X-bqo" which is preserved by products.] §3. Examples ...................................................................................................................... 201 §4. More information ........................................................................................................ -
Ordered Sets
Ordered Sets Motivation When we build theories in political science, we assume that political agents seek the best element in an appropriate set of feasible alternatives. Voters choose their favorite candidate from those listed on the ballot. Citizens may choose between exit and voice when unsatisfied with their government. Consequently, political science theory requires that we can rank alternatives and identify the best element in various sets of choices. Sets whose elements can be ranked are called ordered sets. They are the subject of this lecture. Relations Given two sets A and B, a binary relation is a subset R ⊂ A × B. We use the notation (a; b) 2 R or more often aRb to denote the relation R holding for an ordered pair (a; b). This is read \a is in the relation R to b." If R ⊂ A × A, we say that R is a relation on A. Example. Let A = fAustin, Des Moines, Harrisburgg and let B = fTexas, Iowa, Pennsylvaniag. Then the relation R = f(Austin, Texas), (Des Moines, Iowa), (Harrisburg, Pennsylvania)g expresses the relation \is the capital of." Example. Let A = fa; b; cg. The relation R = f(a; b); (a; c); (b; c)g expresses the relation \occurs earlier in the alphabet than." We read aRb as \a occurs earlier in the alphabet than b." 1 Properties of Binary Relations A relation R on a nonempty set X is reflexive if xRx for each x 2 X complete if xRy or yRx for all x; y 2 X symmetric if for any x; y 2 X, xRy implies yRx antisymmetric if for any x; y 2 X, xRy and yRx imply x = y transitive if xRy and yRz imply xRz for any x; y; z 2 X Any relation which is reflexive and transitive is called a preorder. -
(Pre-)Algebras for Linguistics 1
(Pre-)Algebras for Linguistics 1. Review of Preorders Carl Pollard Linguistics 680: Formal Foundations Autumn 2010 Carl Pollard (Pre-)Algebras for Linguistics An antisymmetric preorder is called an order. The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as `less than or equal to'. (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation v (`less than or equivalent to') on A which is reflexive and transitive. Carl Pollard (Pre-)Algebras for Linguistics The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as `less than or equal to'. (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation v (`less than or equivalent to') on A which is reflexive and transitive. An antisymmetric preorder is called an order. Carl Pollard (Pre-)Algebras for Linguistics If v is an order, then ≡ is just the identity relation on A, and correspondingly v is read as `less than or equal to'. (Pre-)Orders and Induced Equivalence A preorder on a set A is a binary relation v (`less than or equivalent to') on A which is reflexive and transitive. An antisymmetric preorder is called an order. The equivalence relation ≡ induced by the preorder is defined by a ≡ b iff a v b and b v a. -
TAXOTOPY THEORY of POSETS I: VAN KAMPEN THEOREMS 3 of the Category Are Unique
TAXOTOPY THEORY OF POSETS I: VAN KAMPEN THEOREMS AMIT KUBER AND DAVID WILDING Abstract. Given functors F,G ∶C→D between small categories, when is it possible to say that F can be “continuously deformed” into G in a manner that is not necessarily reversible? In an attempt to answer this question in purely category-theoretic language, we use adjunctions to define a ‘taxotopy’ preorder ⪯ on the set of functors C →D, and combine this data into a ‘fundamental poset’ (Λ(C, D), ⪯). The main objects of study in this paper are the fundamental posets Λ(1,P ) and Λ(Z,P ) for a poset P , where 1 is the singleton poset and Z is the ordered set of integers; they encode the data about taxotopy of points and chains of P respectively. Borrowing intuition from homotopy theory, we show that a suitable cone construction produces ‘null-taxotopic’ posets and prove two forms of van Kampen theorem for computing fundamental posets via covers of posets. 1. Introduction and Motivation Let Cat, Pos and Pre denote the categories of small categories and functors, posets and monotone maps, and preorders and monotone maps respectively. Also let Top, SimpComp denote the category of topological spaces and continuous func- tions, simplicial complexes and simplicial maps respectively. One can think of a small category as a topological space via the classifying space functor B ∶ Cat → Top (see [15, §IV.3]), and thus adjectives like connected, homotopy equivalent etc., attributed to topological spaces can also be used for categories. For the full subcategory Pos of Cat, this classifying space functor factors through SimpComp, for given P ∈ Pos, the space BP is the geometric realization of the order complex, ∆(P ), which is a simplicial complex. -
Arxiv:1111.1390V1
On extension of partial orders to total preorders with prescribed symmetric part1 Dmitry V. Akopian and Valentin V. Gorokhovik Institute of Mathematics, The National Academy of Sciences of Belarus, Surganova st., 11, Minsk 220072, Belarus e-mail: [email protected] Abstract For a partial order on a set X and an equivalency relation S defined on the same set X we derive a necessary and sufficient condition for the existence of such a total preorder on X whose asymmetric part contains the asymmetric part of the given partial order and whose symmetric part coincides with the given equivalence relation S. This result generalizes the classical Szpilrajn theorem on extension of a partial order to a perfect (linear) order. Key words partial order, preorder, extension, Szpilrajn theorem, Dushnik-Miller theorem. MSC 2010 Primary: 06A06 Secondary: 06A05, 91B08 Let X be an arbitrary nonempty set and let G ⊂ X × X be a binary relation on X. A binary relation G ⊂ X × X is called a preorder if it is reflexive ( (x, x) ∈ G ∀ x ∈ X ) and transitive ( (x, y) ∈ G, (y, z) ∈ G ⇒ (x, z) ∈ G ∀ x,y,z ∈ X ). If in addition a preorder G ∈ X × X is antisymmetric ( (x, y) ∈ G, (y, x) ∈ G ⇒ x = y ∀ x,y ∈ X ), then it is called a partial order. A total partial order is called a perfect (or linear) order. (A binary relation G ⊂ X × X is total if for any x, y ∈ X either (x, y) ∈ G or (y, x) ∈ G holds.) In the sequel, a preorder will be preferably denoted by the symbol - whereas a partial order as well as a perfect order by the symbol . -
1. the Zermelo Fraenkel Axioms of Set Theory
1. The Zermelo Fraenkel Axioms of Set Theory The naive definition of a set as a collection of objects is unsatisfactory: The objects within a set may themselves be sets, whose elements are also sets, etc. This leads to an infinite regression. Should this be allowed? If the answer is “yes”, then such a set certainly would not meet our intuitive expectations of a set. In particular, a set for which A A holds contradicts our intuition about a set. This could be formulated as a first∈ axiom towards a formal approach towards sets. It will be a later, and not very essential axiom. Not essential because A A will not lead to a logical contradiction. This is different from Russel’s paradox:∈ Let us assume that there is something like a universe of all sets. Given the property A/ A it should define a set R of all sets for which this property holds. Thus R = A∈ A/ A . Is R such a set A? Of course, we must be able to ask this { | ∈ } question. Notice, if R is one of the A0s it must satisfy the condition R/ R. But by the very definition of R, namely R = A A/ A we get R R iff R/∈ R. Of course, this is a logical contradiction. But{ how| can∈ } we resolve it?∈ The answer∈ will be given by an axiom that restricts the definition of sets by properties. The scope of a property P (A), like P (A) A/ A, must itself be a set, say B. Thus, instead of R = A A/ A we define≡ for any∈ set B the set R(B) = A A B, A / A . -
SET THEORY 1. the Suslin Problem 1.1. the Suslin Hypothesis. Recall
SET THEORY MOSHE KAMENSKY Abstract. Notes on set theory, mainly forcing. The first four sections closely follow the lecture notes Williams [8] and the book Kunen [4]. The last section covers topics from various sources, as indicated there. Hopefully, all errors are mine. 1. The Suslin problem 1.1. The Suslin hypothesis. Recall that R is the unique dense, complete and separable order without endpoints (up to isomorphism). It follows from the sepa- rability that any collection of pairwise disjoint open intervals is countable. Definition 1.1.1. A linear order satisfies the countable chain condition (ccc) if any collection of pairwise disjoint open intervals is countable. Hypothesis 1.1.2 (The Suslin hypothesis). (R; <) is the unique complete dense linear order without endpoints that satisfies the countable chain condition. We will not prove the Suslin hypothesis (this is a theorem). Instead, we will reformulate it in various ways. First, we have the following apparent generalisation of the Suslin hypothesis. Theorem 1.1.3. The following are equivalent: (1) The Suslin hypothesis (2) Any ccc linear order is separable Proof. Let (X; <) be a ccc linear order that is not separable. Assume first that it is dense. Then we may assume it has no end points (by dropping them). The completion again has the ccc, so by the Suslin hypothesis is isomorphic to R. Hence we may assume that X ⊆ R, but then X is separable. It remains to produce a dense counterexample from a general one. Define x ∼ y if both (x; y) and (y; x) are separable. This is clearly a convex equivalence relation, so the quotient Y is again a linear order satisfying the ccc.