DOCSLIB.ORG

Partially Ordered Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Partially Ordered Sets Appendix A Partially Ordered Sets Partially ordered sets and complete lattices play a crucial role in program analysis and in this appendix we shall summarise some of their properties. We review the basic approaches for how to construct complete lattices from other complete lattices and state the central properties of partially ordered sets satisfying the Ascending Chain and Descending Chain Conditions. We then review the classical results about least and greatest fixed points. A.l Basic Definitions Partially ordered set. A partial ordering is a relation ~: L x L ---+ { true,fa.lse} that is reflexive (i.e. Vl : l ~ l), transitive (i.e. Vh, l2, b : h ~ l2 A l2 ~ l3 ==> h ~ la), and anti-symmetric (i.e. Vh, h : h ~ l2 Ah ~ h ==> h = l2). A partially ordered set (L, ~) is a set L equipped with a partial ordering ~ (sometimes written ~L). We shall write h ;;;;) h for h ~ l2 and h C l2 for h ~ h A h =1- l2. A subset Y of L has l E L as an upper bound if Vl' E Y : l' ~ l and as a lower bound if Vl' E Y : l' ;;;;) l. A least upper bound l of Y is an upper bound of Y that satisfies l ~ lo whenever lo is another upper bound of Y; similarly, a greatest lower bound l of Y is a lower bound of Y that satisfies lo ~ l whenever lo is another lower bound of Y. Note that subsets Y of a partially ordered set L need not have least upper bounds nor greatest lower bounds but when they exist they are unique (since ~ is anti-symmetric) and they are denoted UY and nY, respectively. Sometimes U is called the join operator and n the meet operator and we shan write h u t2 for U{h, t2} and similarly h n b for n{h, b}. Complete lattice. A complete lattice L = (L, ~) = (L, ~. U, n, _1_, T) is a partially ordered set (L, ~) such that all subsets have least upper bounds 394 A Partially Ordered Sets {2,3} {3} {3} {2,3} (a) Figure A.l: Two complete lattices. as well as greatest lower bounds. Furthermore, _l = U0 = nL is the least element and T = n0 = U L is the greatest element. Example A.l If L = (P(S), ~)for some set S then ~ is ~ and UY = UY, nY = nY, j_ = 0 and T = s. If L = (P(S), 2) then ~ is 2 and UY = nY, nY = UY, j_ = s and T = 0. Hence (P(S), ~) as well as (P(S), 2) are complete lattices. In the case where S = {1, 2, 3} the two complete lattices are shown on Figure A.l; these draw­ ings are often called Hasse diagrams. Here a line "going upwards" from some h to some l2 means that h ~ l2; we do not draw lines that follow from reflexivity or transitivity of the partial ordering. • Lemma A.2 For a partially ordered set L = (L, ~) the claims (i) L is a complete lattice, (ii) every subset of L has a least upper bound, and (iii) every subset of L has a greatest lower bound are equivalent. • Proof Clearly (i) implies (ii) and (iii). To show that (ii) implies (i) let Y ~ L and define (A.l) and let us prove that this indeed defines a greatest lower bound. All the elements of the set on the right hand side of (A.l) are lower bounds of Y so clearly (A.l) defines a lower bound of Y. Since any lower bound of Y will be in the set it follows that (A.l) defines the greatest lower bound of Y. Thus (i) holds. A.l Basic Definitions 395 To show that (iii) implies (i) we define UY = n{l E L 1 Vl' E y : l' ~ l}. Arguments analogous to those above show that this defines a least upper bound and that (i) holds. • Moore family. A Moore family is a subset Y of a complete lattice L = (L, ~) that is closed under greatest lower bounds: VY' ~ Y: nY' E Y. It follows that a Moore family always contains a least element, nY, and a greatest element, n0, which equals the greatest element, T, from L; in particular, a Moore family is never empty. Example A.3 Consider the complete lattice (P(S), ~)of Figure A.l (a). The subsets {{2}, {1, 2}, {2, 3}, {1, 2, 3}} and {0, {1, 2, 3}} are both Moore families, whereas neither of { {1 }, {2}} and {0, {1 }, {2}, {1, 2}} are. • Properties of functions. A function f : L1 ~ L 2 between partially ordered sets L1 = (L1, ~1) and L2 = (L2, ~2) is surjective (or onto or epic) if and it is injective ( or 1-1 or manie) if Vl,l' E L1: f(l) = f(l')::::} l = l' The function f is monotone (or isotone or order-preserving) if \Il, l' E L1 : l ~1 l' ::::} f(l) ~2 f(l') It is an additive function (or a join morphism, sometimes called a distributive function) if and it is called a multiplicative function (ora meet morphism) if The function f is a completely additive function (or a complete join mor­ phism) if for all Y ~ L1: 396 A Partially Ordered Sets and it is completely multiplicative (or a complete meet morphism) if for all Y ~ L1: f(nlY) = n2{f(l') Il' E Y} whenever nly exists Clearly U1 Y and n1Y always exist when L1 is a complete lattice; when L2 is not a complete lattice the above statements also require the appropriate least upper bounds and greatest lower bounds to exist in L2 . The function f is affine if for all non-empty Y ~ L1 J(U lY) = u2{f(l') Il' E Y} whenever uly exists (and y-=!= 0) and it is strict if f(l_l) = 1_ 2 ; note that a function is completely additive if and only if it is both affine and strict. Lemma A.4 If L = (L, !;::;, u, n, 1_, T) and M = (M, !;::;, u, n, 1_, T) are complete lattices and M is finite then the three conditions (i) 'Y : M --+ L is monotone, (ii) 'Y(T) = T, and (iii) 1(m1 n m2) = 'Y(ml) n 1(m2) whenever m 1 g m2 1\ m2 g m 1 are jointly equivalent to 'Y : M --+ L being completely multiplicative. • Proof First note that if 'Y is completely multiplicative then (i), (ii) and (iii) hold. For the converse note that by monotonicity of"'( we have 7(m1nm2) = 1(m1)n"Y(m2) also when m1 [;;:; m2 V m2 [;;:; m1. We then prove by induction on the (finite) cardinality of M' ~ M that: (A.2) If the cardinality of M' is O then (A.2) follows from (ii). If the cardinality of M' is larger than O then we write M' = M" U {m"} where m" f:_ M"; this ensures that the cardinality of M" is strictly less than that of M'; hen ce: "Y((IlM") n m") "Y(I-IM") n 1(m") (1-1 {"Y(m) 1 m E M"}) n "Y(m") 1-lh(m) 1 m E M'} This proves the result. • Lemma A.5 A function f : (P(D), ~) --+ (P(E), ~) is affine if and only if there exists a function 'P : D --+ P(E) and an element 'P0 E P(E) such that f(Y) = U{'P(d) 1 dE Y} U 'P0 The function f is completely additive if and only if additionally 'P0 = 0. • A.2 Construction of Complete Lattices 397 Proof Suppose that f is of the form displayed and let Y be a non-empty set; Then UU(Y) 1 y E Y} u{u{'P(d) 1 dE Y} U 'P0 1 Y E Y} U{U{'P(d) 1 dE Y} 1 Y E Y} U 'P0 u{'P(d) 1 dE uY} u 'P0 f(UY) showing that f is affine. Next suppose that fis affine and define 'P(d) = f({d}) and 'P0 = !(0). For Y E P(D) let Y = {{d} 1 dE Y} U {0} and note that Y = UY and Y =10. Then J(Y) J(UY) u({f({d}) 1 dE Y} U {!(0)}) u({'P(d)) 1 dE Y} U {'P0}) u{'P(d) 1 dE Y} U 'P0 so f can be written in the required form. The additional statement about com­ pletely additivity is straightforward. • An isomorphism from a partially ordered set (L1, ~1) to a partially ordered set (L2, ~ 2 ) is a monotone function (} : L1 -t L2 such that there exists a (necessarily unique) monotone function o- 1 : L2 -t L1 with (}o o- 1 = id2 and o- 1 o(}= id1 (where idi is the identity function over Li, i = 1, 2). A.2 Construction of Complete Lattices Complete lattices can be combined to construct new complete lattices. We shall first see how to construct products and then two kinds of function spaces. Cartesian product. Let L1 = (L1. ~1) and L2 = (L2, ~2) be partially ordered sets. Define L = (L, ~) by and (ln, l21) ~ (h2, l22) iff ln ~1 h2 1\ h1 ~2 b It is then straightforward to verify that L is a partially ordered set. If ad­ ditionally each Li = (Li, ~i, Ui, ni, .li, Ti) is a complete lattice then so is L = (L, ~, U,n, .l, T) and furthermore 398 A Partially Ordered Sets and ..l = (..l1, ..l2) and similarly for nY and T. We often write L1 x L2 for L and call it the cartesian product of L1 and L2 .
Load more

Recommended publications
  • Chapter 4. Homomorphisms and Isomorphisms of Groups
    Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)).
    [Show full text]
  • Scott Spaces and the Dcpo Category
    SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology.
    [Show full text]
  • CTL Model Checking
    CTL Model Checking Prof. P.H. Schmitt Institut fur¨ Theoretische Informatik Fakult¨at fur¨ Informatik Universit¨at Karlsruhe (TH) Formal Systems II Prof. P.H. Schmitt CTLMC Summer 2009 1 / 26 Fixed Point Theory Prof. P.H. Schmitt CTLMC Summer 2009 2 / 26 Fixed Points Definition Definition Let f : P(G) → P(G) be a set valued function and Z a subset of G. 1. Z is called a fixed point of f if f(Z) = Z . 2. Z is called the least fixed point of f is Z is a fixed point and for all other fixed points U of f the relation Z ⊆ U is true. 3. Z is called the greatest fixed point of f is Z is a fixed point and for all other fixed points U of f the relation U ⊆ Z is true. Prof. P.H. Schmitt CTLMC Summer 2009 3 / 26 Finite Fixed Point Lemma Let G be an arbitrary set, Let P(G) denote the power set of a set G. A function f : P(G) → P(G) is called monotone of for all X, Y ⊆ G X ⊆ Y ⇒ f(X) ⊆ f(Y ) Let f : P(G) → P(G) be a monotone function on a finite set G. 1. There is a least and a greatest fixed point of f. S n 2. n≥1 f (∅) is the least fixed point of f. T n 3. n≥1 f (G) is the greatest fixed point of f. Prof. P.H. Schmitt CTLMC Summer 2009 4 / 26 Proof of part (2) of the finite fixed point Lemma Monotonicity of f yields ∅ ⊆ f(∅) ⊆ f 2(∅) ⊆ ..
    [Show full text]
  • Topological Duality and Lattice Expansions Part I: a Topological Construction of Canonical Extensions
    TOPOLOGICAL DUALITY AND LATTICE EXPANSIONS PART I: A TOPOLOGICAL CONSTRUCTION OF CANONICAL EXTENSIONS M. ANDREW MOSHIER AND PETER JIPSEN 1. INTRODUCTION The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper. Regarding objective (a), consider the following simple question: Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing Top. The earliest examples are of the former sort. Tarski [Tar29] (treated in English, e.g., in [BD74]) showed that every complete atomic Boolean lattice is represented by a powerset. Taking some historical license, we can say this result shows that the category of complete atomic Boolean lattices with complete lat- tice homomorphisms is dually equivalent to the category of discrete topological spaces. Birkhoff [Bir37] showed that every finite distributive lattice is represented by the lower sets of a finite partial order. Again, we can now say that this shows that the category of finite distributive lattices is dually equivalent to the category of finite T0 spaces and con- tinuous maps.
    [Show full text]
  • Nieuw Archief Voor Wiskunde
    Nieuw Archief voor Wiskunde Boekbespreking Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046 Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121 Reviewed by Pieter Moree These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1.
    [Show full text]
  • Categories and Types for Axiomatic Domain Theory Adam Eppendahl
    Department of Computer Science Research Report No. RR-03-05 ISSN 1470-5559 December 2003 Categories and Types for Axiomatic Domain Theory Adam Eppendahl 1 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Submitted for the degree of Doctor of Philosophy University of London 2003 2 3 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Abstract Domain Theory provides a denotational semantics for programming languages and calculi con- taining fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dual- sequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to en- richment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which we consider the core types of such calculi, together with the adjoint types. We use the idioms of these typings, which include oblique function spaces, to give a translation of the core of Levy’s Call-By-Push-Value.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • Generic Filters in Partially Ordered Sets Esfandiar Eslami Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1982 Generic filters in partially ordered sets Esfandiar Eslami Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Eslami, Esfandiar, "Generic filters in partially ordered sets " (1982). Retrospective Theses and Dissertations. 7038. https://lib.dr.iastate.edu/rtd/7038 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While most advanced technological means to photograph and reproduce this docum. have been used, the quality is heavily dependent upon the quality of the mate submitted. The following explanation of techniques is provided to help you underst markings or notations which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the documen photographed is "Missing Page(s)". If it was possible to obtain the missin; page(s) or section, they are spliced into the film along with adjacent pages This may have necessitated cutting through an image and duplicatin; adjacent pages to assure you of complete continuity. 2. When an image on the film is obliterated with a round black mark it is ai indication that the film inspector noticed either blurred copy because o movement during exposure, or duplicate copy.
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Direct Products and Homomorphisms
    Commuting properties Direct products and homomorphisms Simion Breaz logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Outline 1 Commuting properties Products and coproducts Contravariant functors Covariant functors logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits. logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits.
    [Show full text]
  • Topology, Domain Theory and Theoretical Computer Science
    Draft URL httpwwwmathtulaneedumwmtopandcspsgz pages Top ology Domain Theory and Theoretical Computer Science Michael W Mislove Department of Mathematics Tulane University New Orleans LA Abstract In this pap er we survey the use of ordertheoretic top ology in theoretical computer science with an emphasis on applications of domain theory Our fo cus is on the uses of ordertheoretic top ology in programming language semantics and on problems of p otential interest to top ologists that stem from concerns that semantics generates Keywords Domain theory Scott top ology p ower domains untyp ed lamb da calculus Subject Classication BFBNQ Intro duction Top ology has proved to b e an essential to ol for certain asp ects of theoretical computer science Conversely the problems that arise in the computational setting have provided new and interesting stimuli for top ology These prob lems also have increased the interaction b etween top ology and related areas of mathematics such as order theory and top ological algebra In this pap er we outline some of these interactions b etween top ology and theoretical computer science fo cusing on those asp ects that have b een most useful to one particular area of theoretical computation denotational semantics This pap er b egan with the goal of highlighting how the interaction of order and top ology plays a fundamental role in programming semantics and related areas It also started with the viewp oint that there are many purely top o logical notions that are useful in theoretical computer science that
    [Show full text]
  • Lecture 1.3: Direct Products and Sums
    Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise.
    [Show full text]