DOCSLIB.ORG

Kashiwa-Nakamura-Convex Gemetries AAB 2012.PDF

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Kashiwa-Nakamura-Convex Gemetries AAB 2012.PDF The rooted circuits and the rooted cocircuits of convex geometries, closure operators, and monotone extensive operators Kashiwabara Kenji and Nakamura Masataka Department of System Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153‐8902, JAPAN. 1 Outline 1. Closure spaces, closure systems, matroids, and convex geometries 2. Supersolvable antimatroids 3. Lattice-embedding of convex geometries to convex geometries 4. Rooted circuits and implicational systems 5. Extensive and intensive operators 6. Relations of areas to convex geometries 2 1. Closure spaces, closure systems, matroids, and convex geometries Let E be a non‐empty finite set. Def. Closure operator :2E 2E A (A) (extensive) A B (A) (B)(monotone) ((A)) (A) (idempotent) Def. (, E) is called a closure space. 3 Def. Closure system Def. Closure operator E E K 2E :2 2 ・EK, ・A(A) ・X,Y K X Y K ・A B (A) (B) ・((A)) (A) A member of a closure system is called a closed set. (A) X K{AE: (A)A} XK,AX one-to-one correspondence 4 Ex. A closure system is a lattice under inclusion relation. {1,2,3,4,5} {1,2,4,5} {2,3,4,5} {1,2,3,5} {2,4,5} {1,2,5} {2,3,5} {1,5} {3,5} {2,5} {5} 5 Closure spaces (closure systems) Def. For each element e E,suppose X E e and X is minimal with respect to the property e (X). Then we say (X,e) is a rooted circuit of a closure operator , (or more generally a monotone extensive operator ). X is called the stem, and e is called the root. We denote the collection of the rooted circuits of by C(). Ex. Rooted circuits {x,y,z} ({y,z}, x) ({x,z}, {x} {y} {z} ({x,y},y) z) 6 Closure spaces (closure systems) Conversely, the rooted-circuit system C ( ) determines the closure operator . (A) A{eE A:(X,e)C(), X A} (A E). If (A)contains an element e E A here is a minimal A such that X A e (X) wirh (X,e) C. e X 7 Closure spaces (closure systems) Def. Rooted cocircuit For e E and Y E e , if Y is minimal with respect to the property e (E(Y e) ), (Y,e) is said to be a rooted cocircuit. We denote the collection of the rooted cocircuits by D(). Ex. {x,y,z} Rooted Cocircuits ({y}, x) ({z}, x) {x} {y} {z} ({z}, y) ({x}, y) ({x}, z) ({y}, z) a (E {y, x}) ({z}){z} 8 Ex. Affine-point configuration E {a,b,c,e} R2 (A) conv.hull(A) E Rooted Circuits a ({a,b,c},e) e b c Rooted Cocircuits ({a},e), ({b},e), ({c},e). 9 Closure spaces (closure systems) Ex. Affine-point configuration E {a,b,c,d,e} R2 Rooted Circuits (A) conv.hull(A) E ({a,b,c},e), ({c,d},e), a ({a,b},d). d e Rooted Cocircuits ({c},e), ({a,d},e), b c ({b,d},e). ({a},d), ({b},d) 10 Closure spaces (closure systems) Def. (A) {eA:e(Ae)} (A E) Is the extreme-point operator of a closure operator . Ex. Affine-point configuration E {a,b,c,d,e} R2. a ({a,b,c,d,e}){a,b,c} d ({a,c,d,e}){a,c,d} e b c ({a,b,e}){a,b,e} 11 Matroid Def. A closure space (,E) A is a matroid if b a(Ab) b(Aa) (Exchange Property) a Abstraction of linear dependency 12 Matroid a(Ab) b(Aa) (Exchange Property) Ex. A graphic matroid b b x x w y w a y a u z u z A{x, y,z} (A) {x, y,z,u,w} 13 Ex. A graphic matroid M(G) of a graph G E=E(G) b Rooted circuits with the root z x ({a, b, x, u}, z), y w a ({x, w, u}, z), ({w, y}, z), u z ({a, b, y}, z). Proposition In a matroid M, for any circuit C and any element eC, (Ce, e) is a rooted circuit, and vice versa. 14 Convex geometry Def. A closure space (, E) is a convex geometry if the corresponding closure operator satisfies the anti‐exchange property. a b, a,b(A), b(Aa) a(Ab) Ex. Affine‐point configuration E R2. (Anti‐exchange Property) (A) conv.hull (A) E (A) b a Abstraction of convexity (Jamison 1982) a(Ab) b(Aa) (Excange Property) 15 Convex geometry Theorem ( Edelman and Jamison 1985) Let (, E) be a closure space , K be the closure system associated with it, and be the extreme‐point operator. Then the following are equivalent. (1) is a closure operator with the anti‐exchange property. (2) For any closed set X K with X E, there exists e E X such that X e K. (3) ( (A)) (A) for A E. (4) ( (A)) (A) for A E. [ Krein‐Milman property] 16 Convex Geometries and Antimatroids Def. Convex geometry Def. Antimatroid K 2E F 2E ,EK, ・ٛ F・ ・X,Y K X Y K ・A,BF ABF ・X K, X E ・AF, A eE X : X eK eA: AeF Def. Antimatroid An element of K is called a F {E X : X K} convex set, and an element of F is a feasible set. 17 E X e A X Ae convex geometry antimatroid 18 {a,b.c,b} {a,b,c,d} {b,c,d} {a,c,d} {a,b,d} {c,d} {a,b} {c} {a} {b} A convex An antimatroid geometry 19 classes of convex geometries ・ Affine convex geometry ・ poset convex geometry ・ double shelling of a poset ・ simplicial shelling of a chordal graph ・ tree node‐shelling convex geometry ・ graph search convex geometry generalized affine convex geometries = all the convex geometries 20 Principle of shelling process of antimatroids Once an element is removable, it remains removable until it is deleted. (A) is the set of removable elements in A. A : E ; while( (A) is non-empty) do delete any element in (A) from A end 21 Ex. Poset double shelling antimatroid – Repeating the deletion of a minimal or maximal element. c {a,b,c,d} d {a,b,c,d} a b {a,c,d} {a,c} {a,b,d} {a,b,c} {a,c,d} {b,c,d} c {a} {a,b} {a,c} {b,c} {c,d} d a b {a} {b} {c} A ｐoset double shelling antimatroid 22 Ex. poset convex geometries c {a,b,c,d} {a,b.c,b} d {a,b,d} {b,c,d} {a,c,d} a b {c,d} {a,b} Hasse diagram of a poset P {a} {b} {c} Deleting a minimal antimatroid convex geometry element of P of P antimatroid 23 Def. Monophonically convex sets of a chordal graph G : a graph, V(G): the vertex set of G X V(G) is monophonically convex if it holds that a,b X and c V is on a chordless path between a and b c X Theorem (Edelman and Jamison 1985) The collection of monophonically convex sets is a convex geometry if and only if G is a chordal graph. 24 Ex. Monophonically convex sets of a chordal graph 125 {12345} {2345} {1345} 3 4 {245} {345} {135} A chordal graph G {24}{25} {45} {34} {35} {15} {13} {2} {4} {5} {3} {1} Removable elements =Simplicial vertex A convex geometry of the chordal graph G 25 Antimatroids and rooted cocircuits Def. Let e E . In an antimatroid F , suppose that an element H F contains e and is minimal with respect to this property. Then we say that ( Y , e ) is a rooted cocircuit where Y H e . e is called the root and Y is the costem. [Remark] H is necessarily a join-irreducible element in the lattice F . 26 Ex. Poset double shelling antimatroid – Repeating the deletion of a minimal or maximal element. {a,b,c,d} c {a,b,d} {a,b,c} {a,c,d} {b,c,d} {a,b} {a,c} {b,c} {c,d} d a b {a} {b} {c} antimatroid Element: minimal feasible set Rooted cocircuits a {a} ( , a), b {b} ( , b), c {c} ( , c) d {a, b, d}, {c, d}, ({a, b}, d), ({c}, d) 27 clutter Def. A clutter is a collection of sets in which no member contains another properly. Def. For a clutter L on E, a subset of E is a transversal of L if it intersects every member of L . The collection of the minimal transversals called the blocker of L and denoted by b ( L ). [Note] If a set intersects every member of a clutter L , then it contains a member of the blocker b(L). 28 clutter Ex. clutters and their blockers L {{a, c},{b, c}}, b (L) {{c},{a, b}}, b(b(L)) L L {{a, b},{b, c},{c,a}}, b (L) {{a,b},{b,c},{c,a}}, b(b(L)) L Proposition For any clutter L, b (b (L)) L holds. 29 Rooted circuits: root stems costems ({a,c}, d), ({b,c}, d) a ー b ー Rooted cocircuits: c ー ( , a), ( , b), ( , c) d {a,c}, {b,c} {a,b}, {c} ({a,b}, d), ({c}, d) L={ {a, c}, {b, c} }, b(L)={ {c}, {a, b} } Proposition (Korte, Lovasz and Schrader) For a convex geometry (K,E) , let C be the collection of the rooted circuits, and D be the collection of the rooted cocircuits. Then for each eE, C(e) {X :(X,e)C}and D(e) {Y :(Y ,e)D}are the blocker of each other.
Load more

Recommended publications
  • Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10
    CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex.
    [Show full text]
  • Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions
    Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions Section 10. 1, Problems: 1, 2, 3, 4, 10, 11, 29, 36, 37 (fifth edition); Section 11.1, Problems: 1, 2, 5, 6, 12, 13, 31, 40, 41 (sixth edition) The notation ""forOR is bad and misleading. Just think that in the context of boolean functions, the author uses instead of ∨.The integers modulo 2, that is ℤ2 0,1, have an addition where 1 1 0 while 1 ∨ 1 1. AsetA is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S,it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers ℕ,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L.
    [Show full text]
  • On Some Open Problems in Algebraic Logic
    On Completions, neat atom structures, and omitting types Tarek Sayed Ahmed Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. October 16, 2018 Abstract . This paper has a survey character, but it also contains several new results. The paper tries to give a panoramic picture of the recent developments in algebraic logic. We take a long magical tour in algebraic logic starting from classical notions due to Henkin Monk and Tarski like neat embeddings, culminating in presenting sophisticated model theoretic constructions based on graphs, to solve problems on neat reducts. We investigate several algebraic notions that apply to varieties of boolean al- gebras with operators in general BAOs, like canonicity, atom-canonicity and com- pletions. We also show that in certain significant special cases, when we have a Stone-like representabilty notion, like in cylindric, relation and polyadic algebras such abtract notions turn out intimately related to more concrete notions, like com- plete representability, and the existence of weakly but not srongly representable atom structures. In connection to the multi-dimensional corresponding modal logic, we prove arXiv:1304.1149v1 [math.LO] 3 Apr 2013 several omitting types theorem for the finite n variable fragments of first order logic, the multi-dimensional modal logic corresponding to CAn; the class of cylindric algebras of dimension n. A novelty that occurs here is that theories could be uncountable. Our construc- tions depend on deep model-theoretic results of Shelah. Several results mentioned in [26] without proofs are proved fully here, one such result is: There exists an uncountable atomic algebra in NrnCAω that is not com- pletely representable.
    [Show full text]
  • Complete Topoi Representing Models of Set Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 57 (1992) l-26 North-Holland Complete topoi representing models of set theory Andreas Blass Mathematics Departinent, University of Michigan, Ann Arbor, MI 48109, United States Andre Scedrov Mathematics Department, University of Pennsylvania, Philadelphia, PA 19104, United States Communicated by T. Jech Received 6 February 1991 Abstract Blass, A. and A. Scedrov, Complete topoi representing models of set theory, Annals of Pure and Applied Logic 57 (1992) l-26. By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued, pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos % of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. 1. Introduction We shall be concerned with models of set theory that are extensions of the ordinary universe V of (pure) sets.
    [Show full text]
  • TOPOLOGICAL REPRESENTATIONS of MATROIDS 1. Introduction One
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 2, Pages 427{442 S 0894-0347(02)00413-7 Article electronically published on November 29, 2002 TOPOLOGICAL REPRESENTATIONS OF MATROIDS E. SWARTZ 1. Introduction One of the foundations of oriented matroid theory is the topological represen- tation theorem of Folkman and Lawrence [8]. It says that an oriented (simple) matroid can be realized uniquely as an arrangement of pseudospheres. That there is no similar interpretation for the class of all matroids has been taken for granted. For instance, \A non-coordinatizable matroid of abstract origin may be thought of as a geometric object only in a purely formal way, whereas an oriented matroid may always be thought of as a geometric-topological configuration on the d-sphere (or in projective space)" [3, p. 19]. Our main theorem is that the class of geometric lattices, which is cryptomorphic to the category of simple matroids, is the same as the class of intersection lattices of arrangements of homotopy spheres. The interpretation of a geometric lattice as an arrangement of homotopy spheres is a natural generalization of the Folkman-Lawrence theorem. An oriented matroid realizable over R has a representation with geodesic spheres. Allowing pseudo- spheres, i.e., those which are homeomorphic, but possibly not isometric to the unit sphere, leads to the category of (simple) oriented matroids. If we further relax the conditions on the spheres to only homotopy equivalence to the standard sphere, then we are led to the category of all (simple) matroids. Some of the theory of oriented matroids which only depends on the underly- ing matroid can be extended to homotopy sphere arrangements.
    [Show full text]
  • Lattice Theory Lecture 1 Basics
    Lattice Theory Lecture 1 Basics John Harding New Mexico State University www.math.nmsu.edu/∼JohnHarding.html [email protected] Toulouse, July 2017 References Some useful books on lattice theory Birkhoff: Lattice Theory Crawley and Dilworth: The Algebraic Theory of Lattices Balbes and Dwinger: Distributive Lattices Davey and Priestley: Introduction to Lattices and Order Birkhoff is older and advanced, and dated, but it is still the best. It conveys what lattice theory is | pervasive. The others are all good as companions and to learn details. Gr¨atzeris encyclopedic. 2 / 43 Background The prime feature of lattice theory is its versatility. It connects many areas. Algebra, analysis, topology, logic, computer science, combinatorics, linear algebra, geometry, category theory, probability. We will touch on all these topics. If you know a couple, great, you will learn a little about the others. 3 / 43 Partial orders Definition A partially ordered set or poset, is a set P with a binary relation on P that satisfies for all x; y; z P 1. reflexive:≤ x x ∈ 2. anti-symmetric: x y and y x x y ≤ 3. transitive: x y and y z x z ≤ ≤ ⇒ = If it additionally satisfies≤ ≤ ⇒ ≤ 4. x y or y z the poset≤ is called≤ a liner order or a chain. 4 / 43 Hasse diagrams We draw pictures of finite posets using dots for the elements of P and having x z if there are upward line segments from x to z. ≤ z x We don't include the line from x to z because it is implied by our drawing a picture of a transitive relation.
    [Show full text]
  • Poset Topology: Tools and Applications
    Poset Topology: Tools and Applications Michelle L. Wachs IAS/Park City Mathematics Institute, Summer 2004 Contents Poset Topology: Tools and Applications 1 Introduction 3 Lecture 1. Basic definitions, results, and examples 5 1.1. Order complexes and face posets 5 1.2. The M¨obius function 9 1.3. Hyperplane and subspace arrangements 11 1.4. Some connections with graphs, groups and lattices 16 1.5. Poset homology and cohomology 17 1.6. Top cohomology of the partition lattice 19 Lecture 2. Group actions on posets 23 2.1. Group representations 23 2.2. Representations of the symmetric group 25 2.3. Group actions on poset (co)homology 30 2.4. Symmetric functions, plethysm, and wreath product modules 32 Lecture 3. Shellability and edge labelings 41 3.1. Shellable simplicial complexes 41 3.2. Lexicographic shellability 45 3.3. CL-shellability and Coxeter groups 57 3.4. Rank selection 62 Lecture 4. Recursive techniques 67 4.1. Cohen-Macaulay complexes 67 4.2. Recursive atom orderings 71 4.3. More examples 73 4.4. The Whitney homology technique 77 4.5. Bases for the restricted block size partition posets 82 4.6. Fixed point M¨obius invariant 90 Lecture 5. Poset operations and maps 91 5.1. Operations: Alexander duality and direct product 91 5.2. Quillen fiber lemma 95 5.3. General poset fiber theorems 100 5.4. Fiber theorems and subspace arrangements 103 5.5. Inflations of simplicial complexes 105 Bibliography 109 2 IAS/Park City Mathematics Series Volume 00, 2004 Poset Topology: Tools and Applications Michelle L.
    [Show full text]
  • An Introduction to Boolean Algebras
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 12-2016 AN INTRODUCTION TO BOOLEAN ALGEBRAS Amy Schardijn California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Algebra Commons Recommended Citation Schardijn, Amy, "AN INTRODUCTION TO BOOLEAN ALGEBRAS" (2016). Electronic Theses, Projects, and Dissertations. 421. https://scholarworks.lib.csusb.edu/etd/421 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. An Introduction to Boolean Algebras A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Amy Michiel Schardijn December 2016 An Introduction to Boolean Algebras A Thesis Presented to the Faculty of California State University, San Bernardino by Amy Michiel Schardijn December 2016 Approved by: Dr. Giovanna Llosent, Committee Chair Date Dr. Jeremy Aikin, Committee Member Dr. Corey Dunn, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular nota- tion. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra.
    [Show full text]
  • C Matroid Invariants and Greedoid Invariants 95
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2004 Greedoid invariants and the greedoid Tutte polynomial Chris Anne Clouse The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Clouse, Chris Anne, "Greedoid invariants and the greedoid Tutte polynomial" (2004). Graduate Student Theses, Dissertations, & Professional Papers. 9492. https://scholarworks.umt.edu/etd/9492 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. Maureen and Mike MANSFIELD LIBRARY The University of Montana Permission is granted by the author to reproduce this material in its entirety, provided that this material is used for scholarly purposes and is properly cited in published works and reports. '*Please check "Yes" or "No" and provide signature Yes, I grant permission No, I do not grant permission Author's Signature: Date: 0 H Any copying for commercial purposes or financial gain may be undertaken only with the author's explicit consent. 8/98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Greedoid invariants and the greedoid Tutte polynomial by Chris Anne Clouse presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy The University of Montana May 2004 Approved by: ■person Dean, Graduate School Date Reproduced with permission of the copyright owner.
    [Show full text]
  • Foundations of Applied Mathematics I
    Foundations of Applied Mathematics I Jeffrey Ketland University of Warsaw [email protected] Forthcoming in Synthese November 23, 2020 Abstract This paper aims to study the foundations of applied mathematics, using a for- malized base theory for applied mathematics: ZFCAσ (Zermelo-Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, represen- tation hypotheses, transfer principles and abstract equivalents. Contents 1 Mathematicized Theories2 2 How The Laws of Set Theory Apply8 3 A Base Theory for Applied Mathematics9 4 Comprehension and Set Abstracts 17 5 Abstract Counterparts: Equivalence Modulo Mathematics 20 6 Systems and Equivalences 21 7 Application Conditionals 30 8 Representation and Transfer 34 9 Summary 37 A Four-Sorted System 38 1 1 Mathematicized Theories Standard scientific theory supplies much of the information it supplies about physical entities only indirectly, by way of apparatus pertaining to supposed relationships of physical entities to supposed mathematical entities and sup- posed classifications of and relationships among the supposed mathematical entities themselves. As much of what science says about observable entities is “theory-laden”, so much of what science says about concrete entities (observable or theoretical) is “abstraction-laden”. (Burgess & Rosen(1997): 84) 1.1 The Logical Structure of Applied Mathematics This paper aims to study the foundations of applied mathematics. Perhaps contrary to current fashions, I treat applied mathematics as taking place within a certain formalized system: specifically, Zermelo-Fraenkel set theory (with Choice) with atoms, over an application signature σ.
    [Show full text]
  • NF Is Consistent
    Quine’s “New Foundations” is consistent M. Randall Holmes 1/2/2017 3:30 PM: version to be submitted Contents 1 Introduction (Abstract) 2 2 The simple theory of types 3 3 The definition of New Foundations 6 4 Well-known results about New Foundations 8 5 Consistency of NFU 10 6 Tangled type theories 13 6.1 ω- and α-modelsfromtangledtypetheory . 15 7 Tangled webs of cardinals 17 8 The main construction 19 arXiv:1503.01406v13 [math.LO] 2 Jan 2017 9 Conclusions and questions 53 10 References and Index 55 1 Version of 1/2/2017 3:30 PM Boise time: version to be submitted 2 1 Introduction (Abstract) In this paper we will present a proof of the consistency of Quine’s set theory “New Foundations” (hereinafter NF), so-called after the title of the 1937 paper [11] in which it was introduced. The strategy is to present a Fraenkel- Mostowski construction of a model of an extension of Zermelo set theory without choice whose consistency was shown to entail consistency of NF in our paper [5] of 1995. There is no need to refer to [5]: this paper presents a full (we think a better) account of considerations drawn from that paper. Version of 1/2/2017 3:30 PM Boise time: version to be submitted 3 2 The simple theory of types New Foundations is introduced as a modification of a simple typed theory of sets which we will call “the simple theory of types” and abbreviate TST (following the usage of Thomas Forster and others).
    [Show full text]
  • Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2008 Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras John D. LaGrange University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Algebra Commons Recommended Citation LaGrange, John D., "Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras. " PhD diss., University of Tennessee, 2008. https://trace.tennessee.edu/utk_graddiss/393 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by John D. LaGrange entitled "Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. David F. Anderson, Major Professor We have read this dissertation and recommend its acceptance: Michael Langston, Shashikant Mulay, Pavlos Tzermias Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by John D. LaGrange entitled “Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras.” I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics.
    [Show full text]