DOCSLIB.ORG

Uniform Semimodular Lattice & Valuated Matroid

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Uniform Semimodular Lattice & Valuated Matroid Uniform Semimodular Lattice & Valuated Matroid Hiroshi Hirai University of Tokyo [email protected] JSIAM, Osaka University Osaka, March 15, 2018 1 Contents 1. Matroid v.s. geometric lattice 2. Valuated matroid 3. Result: uniform semimoduler lattice • examples • valuated matroid uniform semimodular lattice • uniform semimodular lattice valuated matroid 4. Concluding remarks 2 Matroid : base family [EXC] , 3 Representable Matroid is a basis } ): matroid 4 Lattice Lattice := partially ordered set s. t. join & meet exist atom 5 Geometric Lattice := atomistic semimodular lattice semimodular: atomistic: { atom } is geometric lattice ordered by 6 Matroid v.s. Geometric Lattice Birkoff 1940 : matroid The family of flats is a geometric lattice. geometric lattice of height , the set of atoms ) is a (simple) matroid 7 Valuated Matroid Dress-Wenzel 1990 Valuated matroid on matroid : • Greedy algorithm • M-convexity, Discrete convex analysis (Murota 1996～) • Tropical Plucker vector (Speyer-Sturmfels 2004, Speyer 2008) 8 Representable Valuated Matroid is a basis } is a valuated matroid [VEXC] = Tropicalization of Grassmann-Plucker identity 9 Tree Metric tree Four-point condition =VEXC is a valuated matroid 10 Result Matroid Geometric Lattice integer-valued Uniform Valuated Matroid Semimodular Lattice 11 Definition Geometric lattice = atomistic semimodular lattice semimodular: atomistic: { atom } Uniform semimodular lattice semimodular: uniform: is automorphism is geometric lattice 12 Example 1: Integer Lattice is uniform (semi)modular lattice = min, = max, 13 Example 2. is uniform (semi)modular lattice corresponding to tree metric 14 Example 3. Lattice of Lattice Lattice = rank free -submodule of ordered by is uniform semimodular lattice corresponding to deg det 15 Valuated matroid Uniform semimodular lattice valuated matroid on Tropical linear space (Dress-Terhalle 1993, Murota-Tamura 2001, Speyer 2008) Tropical convexity (Murota-Tamura 2001,Develin-Sturmfels 2004) automorphism semimodularity is uniform semimodular lattice 16 Uniform semimodular lattice Valuated matroid Geometric lattice matroid on atoms Uniform semimodular lattice valuated matroid on ends Ray: s.t. End: an equivalence class of rays by parallel relation 17 the set of all ends -rays representatives of geometric lattice atoms Matroid at : Matroid at infinity: Valuated matroid on 18 Fix arbitrarily is -sublattice is valuated matroid 19 Concluding remarks • Lattice-characterization to valuated matroid & tropical linear space • = Dress-Terhalle completion Dress-Terhalle 1993 • Modular matroid = Spherical building of type A Birkhoff 1940, Tits 1974 Modular valuated matroid = Euclidean building of type A 20 • L-convex function on uniform modular lattice References H. Hirai: Uniform semimodular lattice and valuated matroid, arXiv, 2018. H. Hirai: Uniform modular lattice and Euclidean building, arXiv, 2017. 21.
Load more

Recommended publications
  • Modular and Semimodular Lattices Ph.D. Dissertation
    Modular and semimodular lattices Ph.D. Dissertation Benedek Skublics Supervisor: Prof. G´abor Cz´edli Doctoral School in Mathematics and Computer Science University of Szeged, Bolyai Institute 2013 Szeged Contents Introduction 1 1 Von Neumann frames 5 1.1 Basic definitions and notions . 7 1.2 The product frame . 12 1.3 The ring of an outer von Neumann frame . 17 1.3.1 A pair of reciprocal mappings . 19 1.3.2 Addition and further lemmas . 22 1.3.3 Multiplication . 23 2 Isometrical embeddings 28 2.1 Motivation: the finite case . 30 2.1.1 Matroids . 32 2.1.2 Embeddings with matroids . 35 2.2 The general case . 38 2.2.1 Basic concepts and lemmas . 39 2.2.2 The main proofs . 45 2.2.3 Examples . 48 3 Mal'cev conditions 50 3.1 Definition of a Mal'cev condition . 52 3.2 Congruences of algebras with constants . 53 Summary 60 Osszefoglal´o¨ 65 Bibliography 70 i Acknowledgment First of all, I would like to express my sincere gratitude to my supervisor, G´abor Cz´edli,who gave me my first research problem and also taught me how to write mathematics. I am thankful for his constant support and patience. He never got tired of correcting the returning mathematical and grammatical mistakes in my manuscripts. I hope I do not bring discredit upon him with this dissertation. I am also grateful to P´eterP´alP´alfy, my former supervisor at ELTE, who sug- gested coming to the Bolyai Insitute. Both his mathematical thinking and his per- sonality meant a lot to me.
    [Show full text]
  • Lecture Notes on Algebraic Combinatorics Jeremy L. Martin
    Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T .
    [Show full text]
  • Consistent Dually Semimodular Lattices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series A 60, 246-263 (1992) Consistent Dually Semimodular Lattices KAREN M. GRAGG AND JOSEPH P.S. KUNG* Department of Mathematics, Universiiy of North Texas, Denton, Texas 76203 Communicated by Gian-Carlo Rota Received July 30, 1990 In this paper, we study combinatorial and order-theoretical properties of consis- tent dually semimodular lattices of finite rank. Natural examples of such lattices are lattices of subnormal subgroups of groups with composition series. We prove that a dually semimodular lattice is consistent if and only if it is “locally” atomic and modular. From this, we conclude that dually semimodular subgroup lattices are necessarily consistent. We discuss exchange properties for irredundant decomposi- tions of elements into join-irreducible% Finally, we show that a finite consistent dually semimodular lattice is modular if and only if it has the same number of join-irreducibles and meet-irreducibles. 0 1992 Academic Press, Inc. 1. INTRODUCTION A lattice L is dually semimodular (or lower semimodular) if for all elements x and y in L, x v y covers x implies y covers x A y. An element j in L is a join-irreducible if j= a v b implies j= a or j= b, or, equivalently, j covers at most one element. A lattice L is consistent if for all join- irreducible j and all elements x in L, j v x is a join-irreducible in the upper interval [Ix, I].
    [Show full text]
  • Join-Semidistributive Lattices and Convex Geometries
    Advances in Mathematics 173 (2003) 1–49 http://www.elsevier.com/locate/aim Join-semidistributive lattices and convex geometries K.V. Adaricheva,*,1 V.A. Gorbunov,2 and V.I. Tumanov Institute of Mathematics of the Siberian Branch of RAS, Acad. Koptyug Prosp. 4, Novosibirsk, 630090 Russia Received 3 June 2001; accepted 16 April 2002 Abstract We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embeddingresults in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice SPðAÞ of algebraic subsets of a suitable algebraic lattice A: This latter construction, SPðAÞ; is a key example of a convex geometry that plays an analogous role in hierarchy of join- semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries. r 2002 Elsevier Science (USA). All rights reserved. MSC: 06B05; 06B15; 08C15; 06B23 Keywords: Lattice; Join-semidistributive; Anti-exchange property; Convex geometry; Atomistic; Biatomic; Quasivariety; Antimatroid A lattice ðL; 3; 4Þ is called join-semidistributive,if x3y ¼ x3z implies that x3y ¼ x3ðy4zÞ for all x; y; zAL: *Correspondingauthor. E-mail address: [email protected] (K.V. Adaricheva). 1 Current address: 134 South Lombard Ave., Oak Park, IL 60302, USA.
    [Show full text]
  • Weakly Submodular Rank Functions, Supermatroids, and the Flat Lattice Of
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 999–1017 www.elsevier.com/locate/disc Weakly submodular rank functions, supermatroids, and the flat lattice of a distributive supermatroid Marcel Wild Department of Mathematics, University of Stellenbosch, Matieland 7602, South Africa Received 20 October 2005; received in revised form 23 October 2006; accepted 19 March 2007 Available online 24 March 2007 In Memoriam Gian-Carlo Rota Abstract Distributive supermatroids generalize matroids to partially ordered sets. Completing earlier work of Barnabei, Nicoletti and Pezzoli we characterize the lattice of flats of a distributive supermatroid. For the prominent special case of a polymatroid the description of the flat lattice is particularly simple. Large portions of the proofs reduce to properties of weakly submodular rank functions. The latter are also investigated for their own sake, and some new results on general supermatroids are derived. © 2007 Elsevier B.V. All rights reserved. Keywords: Distributive supermatroid; Submodular function; Meet subsemilattice; Rank function; Greedoid; Selector; Supermatroid; Polymatroid; Semimodular lattice; Distributive lattice; Join-prime irreducible element; Independent order ideal; Unit increase; Closure space; Exchange property 1. Introduction The correspondence between closure operators on a lattice D and ∧-subsemilattices of D is well known. In Section 2 we review from [18] the correspondence between ∧-subsemilattices and certain weakly submodular rank functions D → N, and indicate how the applicability of weakly submodular rank functions (WSRFs) extends beyond the present article. In Section 3, starting with matroids, we introduce the more general class of Faigle-WSRFs and then the intermediate class of distributive supermatroids (DSMs).
    [Show full text]
  • Uniform Semimodular Lattices and Valuated Matroids
    Uniform semimodular lattices and valuated matroids Hiroshi HIRAI Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan. [email protected] March 4, 2019 Abstract In this paper, we present a lattice-theoretic characterization for valuated ma- troids, which is an extension of the well-known cryptomorphic equivalence be- tween matroids and geometric lattices (= atomistic semimodular lattices). We introduce a class of semimodular lattices, called uniform semimodular lattices, and establish a cryptomorphic equivalence between integer-valued valuated ma- troids and uniform semimodular lattices. Our result includes a coordinate-free lattice-theoretic characterization of integer points in tropical linear spaces, incor- porates the Dress-Terhalle completion process of valuated matroids, and estab- lishes a smooth connection with Euclidean buildings of type A. Keywords: Valuated matroid, uniform semimodular lattice, geometric lattice, trop- ical linear space, tight span, Euclidean building. 1 Introduction Matroids can be characterized by various cryptomorphically equivalent axioms; see e.g., [1]. Among them, a lattice-theoretic characterization by Birkhoff [4] is well-known: The lattice of flats of any matroid is a geometric lattice (= atomistic semimodular lattice), and any geometric lattice gives rise to a simple matroid. The goal of the present article is to extend this classical equivalence to valuated arXiv:1802.08809v2 [math.CO] 1 Mar 2019 matroids (Dress and Wenzel [8, 9]). Valuated matroid is a quantitative generalization of matroid, which abstracts linear dependence structures of vector spaces over a field with a non-Archimedean valuation. A valuated matroid is defined as a function on the base family of a matroid satisfying a certain exchange axiom that originates from the Grassmann-Pl¨ucker identity.
    [Show full text]