DOCSLIB.ORG

MTH 562/662 Spring 2019 Abstract Algebra II Drew Armstrong Contents

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MTH 562/662 Spring 2019 Abstract Algebra II Drew Armstrong Contents MTH 562/662 Spring 2019 Abstract Algebra II Drew Armstrong Contents Week 13 The Classical Problem of Algebra, Definition of Fields and Subfields/Extensions, Adjunction: The Subfield Generated by a Subset 2 Week 14 The Lattice of Subfields, Definition of the Galois Group, Splitting Fields, The Fundamental Theorem of Galois Theory 10 Problem Set 7 22 Week 15 Rings and Subrings/Extensions, Ring Homomorphisms, Ideals and Quotient Rings, Isomorphism Theorems for Rings 31 Week 16 Ideal and Subring Generated by a Subset, Fields are Simple Rings, Division With Remainder, Descartes' Factor Theorem, Z and F[x] are PIDs 38 Problem Set 8 49 Week 17 Integral Domains, Euclidean Domains, Principal Ideal Domains (PIDs) Prime and Irreducible Elements, Unique Factorization Domains (UFDs) 58 Week 18 Evaluation of Polynomials, The Minimal Polynomial Theorem, Kronecker's Theorem, Existence of Splitting Fields 71 Problem Set 9 81 Week 19 Irreducible Polynomials, Rational Root Test, The Splitting Field of x3 − 2, Cyclotomic Polynomials, Constructible Numbers 96 Week 20 Fields of Size Four and Eight, Primitive Root Theorem, Repeated Roots, Frobenius Automorphism, Existence and Uniqueness of Finite Fields 102 Problem Set 10 112 1 Week 21 Multi-Variable Polynomials, The Finiteness Theorem, Definition of Galois Extensions, The Lifting Lemma, The Splitting Field Theorem 123 Week 22 Perfect Fields, Artin's Fixed Field Lemma, Galois Extensions Over Perfect Fields, The Fundamental Theorem of Galois Theory 134 Problem Set 11 144 Epilogue: Galois' Solvability Theorem 149 Week 13 Last semester I used the story of Galois Theory to motivate the study of abstract groups. This semester I will use the same story to motivate the study abstract rings and fields. As before, we will find that linear algebra is always hiding just beneath the surface. To begin, let me refresh your memory. The classical (pre-1830) problem of algebra was to find explicit \formulas" for the roots of a polynomial equation. To be precise, suppose that some rational numbers (called \coefficients") are given: e1; e2; : : : ; en 2 Q: Then we want to find some numbers r1; r2; : : : ; rn (called \roots") such that n n−1 n−2 n x − e1x + e2x − · · · + (−1) en = (x − r1)(x − r2) ··· (x − rn): A priori, it is not obvious what kind of \numbers" the roots should be, or whether they exist at all. Soon we will prove a result called the Fundamental Theorem of Algebra which says that the roots always exist in the field C of complex numbers. Unfortunately, this theorem will not tell us how to find the roots. We would really like to have some formula or algorithm for computing the roots. To state the problem explicitly, we expand the right hand side and then equate coefficients to obtain a system of n non-linear equations in n unknowns: 8 e = r + r + ··· + r > 1 1 2 n > e = r r + r r + ··· + r r > 2 1 2 1 3 n−1 n > . <> . P ek = ri ri ··· ri > 1≤i1<i2<···<ik≤n 1 2 k > . > . > : en = r1r2 ··· rn: 2 Our goal is to somehow \invert" this system. The best we could hope for is to find some n explicit functions f1; f2; : : : ; fn from Q to C such that 8 r1 = f1(e1; e2; : : : ; en) > <> r2 = f2(e1; e2; : : : ; en) . > . > : rn = fn(e1; e2; : : : ; en): But this hope is too naive. Indeed, no such functions can exist. To see why, observe that each coefficient ek can be thought of as a function of the roots: X ek = ek(r1; r2; : : : ; rn) = ri1 ri2 ··· rik : 1≤i1<i2<···<ik≤n Furthermore, this function has the nice property of being \symmetric" under permutations of the roots. In other words, if σ 2 Sn is any permutation of the set f1; 2; : : : ; ng then we have ek(r1; r2; : : : ; rn) = ek(rσ(1); rσ(2); : : : ; rσ(n)): The easiest way to see this is to observe that the product (x − r1)(x − r2) ··· (x − rn) is symmetric under permutations of its factors. If we expand the product then each coefficient must also be a symmetric function. [Jargon. The coefficients e1; : : : ; en are called the elementary symmetric functions of the roots. This explains my use of the letter \e."] n If f : Q ! C is any function then we can think of the expression f(e1; : : : ; en) as a function of the roots, as follows: f(e1; : : : ; en)(r1; : : : ; rn) := f (e1(r1; : : : ; rn); : : : ; en(r1; : : : ; rn)) : Furthermore, it is clear that f(e1; : : : ; en) is a symmetric function of the roots. Now we see why our first hope was too naive: There can be no function fk such that rk = fk(e1; e2; : : : ; en) because fk(e1; e2; : : : ; en) is always a symmetric function of the roots, whereas rk is certainly not a sym- metric function of the roots. The solution is to weaken the requirement that fk is a \function." Instead, we will allow \multi-valued functions"1 such as square roots. We often talk about \the" square root as though it were a function p − : C ! C: p But this is not a function. Indeed, for any 0 6= α 2 C, the expression α represents two distinct complex numbers and there is no natural way to choose between them. We can use this ambiguity to solve the quadratic equation. 1Recall that a \multi-valued function" is not a function. This is a terrible but common notation. 3 Example: The Quadratic Formula. For any rational coefficients e1; e2 2 Q we want to find some complex roots r1; r2 2 C such that 2 2 x − e1x + e2 = (x − r1)(x − r2) = x − (r1 + r2)x + (r1r2): In other words, we want to find some \multi-valued functions" f1; f2 such that e = r + r r = f (e ; e ) 1 1 2 () 1 1 1 2 e2 = r1r2 r2 = f2(e1; e2): As you know, the solution is ( p 2 f1(e1; e2) = (e1 + e1 − 4e2)=2 p 2 f2(e1; e2) = (e1 − e1 − 4e2)=2: p 2 The only subtlety here is that we must interpret the ambiguous expression \ e1 − 4e2" in the p 2 same way for both equations. In other words, we let \ e1 − 4e2" denote one particular 2 2 2 number α 2 C such that α = e1 − 4e2. If e1 − 4e2 6= 0 then there will be two choices and we just pick one at random.2 /// The process of choosing a random square root is called \breaking the symmetry." For higher degree equations we expect that we will need to break the symmetry by choosing random 3rd roots, 4th roots, etc. This leads us to the classical problem of algebra. The Classical Problem of Algebra. Let e1; : : : ; en 2 Q be any rational numbers. By the Fundamental Theorem of Algebra there exist some unique complex numbers r1; : : : ; rn 2 C such that n n−1 n−2 n x − e1x + e2x − · · · + (−1) en = (x − r1)(x − r2) ··· (x − rn): Our goal is to find some way to compute these roots. Specifically, we want to find an \algebraic formula" expressing the roots in terms of the coefficients, using only the \algebraic operations" p p p p +; −; ×; ÷; ; 3 ; 4 ; 5 ;:::: /// As you know, this problem turns out to be impossible when n ≥ 5. Last semester we developed the group theory necessary for the proof of impossibility.3 This semester we will fill in the other half of the proof. 2If the roots are imaginary then there is a deep sense in which they are indistinguishable. Indeed, we usually define the imaginary unit i as \the" square root of −1. But if −1 has any square root then it must have two. Which one do you want to call i ? 3 Specifically, we proved that the group Sn not \solvable" when n ≥ 5. 4 The Classical Problem was definitively solved by Galois in the 1820s. However, he died too soon to really explain it to anyone. Galois' work was eventually published in 1846 by Joseph Liouville. The first textbook on Galois theory was Camille Jordan's Trait´edes substitutions et des ´equationsalg´ebraiques (1870). At this point \Galois theory" and \group theory" were the same subject, so Jordan's work can also be viewed as the first book about groups. However, the subject was still difficult to understand. The next major advance was made by Richard Dedekind in the 1880s when he defined the concept of a field.4 Definition of Fields and Subfields/Extensions. A field is a set F together with two binary operations +; × : F × F ! F (called addition and multiplication) and two special elements 0; 1 2 F (called zero and one), which satisfy the following three axioms: (F1) (F; +; 0) is an abelian group. (F2) (F − f0g; ×; 1) is an abelian group. We will use juxtaposition to denote multiplication: ab := a × b: Furthermore, since multiplication is commutative we are free to use fractional notation to denote division: a ab−1 = b−1a = : b (F3) Distribution. For all a; b; c 2 F we have a(b + c) = ab + ac: Now let S ⊆ F be any subset. We say that S is a subfield of F (equivalently, F is a field extension of S) if the following properties are satisfied: • The special elements 0; 1 are in S. • For all a; b 2 S we have a ± b 2 S. 4Dedekind's name for this structure was K¨orper, short for Zahlk¨orper [body of numbers]. Dedekind's rival Leopold Kronecker used the term Rationalit¨atsbereich [domain of rationality]. The English term field was coined by E. H. Moore in 1893, possibly motivated by the word \domain." This creates a problem for English speakers: should we denote fields by the letter K or the letter F ? To avoid confusion I will use the blackboard bold font (i.e., K or F) to denote fields.
Load more

Recommended publications
  • GALOIS THEORY for ARBITRARY FIELD EXTENSIONS Contents 1
    GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS PETE L. CLARK Contents 1. Introduction 1 1.1. Kaplansky's Galois Connection and Correspondence 1 1.2. Three flavors of Galois extensions 2 1.3. Galois theory for algebraic extensions 3 1.4. Transcendental Extensions 3 2. Galois Connections 4 2.1. The basic formalism 4 2.2. Lattice Properties 5 2.3. Examples 6 2.4. Galois Connections Decorticated (Relations) 8 2.5. Indexed Galois Connections 9 3. Galois Theory of Group Actions 11 3.1. Basic Setup 11 3.2. Normality and Stability 11 3.3. The J -topology and the K-topology 12 4. Return to the Galois Correspondence for Field Extensions 15 4.1. The Artinian Perspective 15 4.2. The Index Calculus 17 4.3. Normality and Stability:::and Normality 18 4.4. Finite Galois Extensions 18 4.5. Algebraic Galois Extensions 19 4.6. The J -topology 22 4.7. The K-topology 22 4.8. When K is algebraically closed 22 5. Three Flavors Revisited 24 5.1. Galois Extensions 24 5.2. Dedekind Extensions 26 5.3. Perfectly Galois Extensions 27 6. Notes 28 References 29 Abstract. 1. Introduction 1.1. Kaplansky's Galois Connection and Correspondence. For an arbitrary field extension K=F , define L = L(K=F ) to be the lattice of 1 2 PETE L. CLARK subextensions L of K=F and H = H(K=F ) to be the lattice of all subgroups H of G = Aut(K=F ). Then we have Φ: L!H;L 7! Aut(K=L) and Ψ: H!F;H 7! KH : For L 2 L, we write c(L) := Ψ(Φ(L)) = KAut(K=L): One immediately verifies: L ⊂ L0 =) c(L) ⊂ c(L0);L ⊂ c(L); c(c(L)) = c(L); these properties assert that L 7! c(L) is a closure operator on the lattice L in the sense of order theory.
    [Show full text]
  • Formal Contexts, Formal Concept Analysis, and Galois Connections
    Formal Contexts, Formal Concept Analysis, and Galois Connections Jeffrey T. Denniston Austin Melton Department of Mathematical Sciences Departments of Computer Science and Mathematical Sciences Kent State University Kent State University Kent, Ohio, USA 44242 Kent, Ohio, USA 44242 [email protected] [email protected] Stephen E. Rodabaugh College of Science, Technology, Engineering, Mathematics (STEM) Youngstown State University Youngstown, OH, USA 44555-3347 [email protected] Formal concept analysis (FCA) is built on a special type of Galois connections called polarities. We present new results in formal concept analysis and in Galois connections by presenting new Galois connection results and then applying these to formal concept analysis. We also approach FCA from the perspective of collections of formal contexts. Usually, when doing FCA, a formal context is fixed. We are interested in comparing formal contexts and asking what criteria should be used when determining when one formal context is better than another formal context. Interestingly, we address this issue by studying sets of polarities. 1 Formal Concept Analysis and Order-Reversing Galois Connections We study formal concept analysis (FCA) from a “larger” perspective than is commonly done. We em- phasize formal contexts. For example, we are interested in questions such as if we are working with a given formal context K , that is, we are working with a set of objects G, a set of properties M, and a relation R ⊂ G×M, what do we do if we want to replace K with a better formal context. Of course, this raises the question: what makes one formal context better than another formal context.
    [Show full text]
  • Logics from Galois Connections
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Journal of Approximate Reasoning 49 (2008) 595–606 Contents lists available at ScienceDirect International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar Logics from Galois connections Jouni Järvinen a,*, Michiro Kondo b, Jari Kortelainen c a Turku Centre for Computer Science (TUCS), University of Turku, FI-20014 Turku, Finland b School of Information Environment, Tokyo Denki University, Inzai 270-1382, Japan c Mikkeli University of Applied Sciences, P.O. Box 181, FI-50101 Mikkeli, Finland article info abstract Article history: In this paper, Information Logic of Galois Connections (ILGC) suited for approximate rea- Received 13 June 2007 soning about knowledge is introduced. In addition to the three classical propositional logic Received in revised form 29 May 2008 axioms and the inference rule of modus ponens, ILGC contains only two auxiliary rules of Accepted 16 June 2008 inference mimicking the performance of Galois connections of lattice theory, and this Available online 27 June 2008 makes ILGC comfortable to use due to the flip-flop property of the modal connectives. Kripke-style semantics based on information relations is defined for ILGC. It is also shown that ILGC is equivalent to the minimal tense logic K , and decidability and completeness of Keywords: t ILGC follow from this observation. Additionally, relationship of ILGC to the so-called clas- Rough sets Fuzzy sets sical modal logics is studied. Namely, a certain composition of Galois connection mappings Approximate reasoning forms a lattice-theoretical interior operator, and this motivates us to axiomatize a logic of Knowledge representation these compositions.
    [Show full text]
  • A Galois Connection Calculus for Abstract Interpretation∗
    A Galois Connection Calculus for Abstract Interpretation⇤ Patrick Cousot Radhia Cousot CIMS⇤⇤, NYU, USA [email protected] CNRS Emeritus, ENS, France s [email protected] rfru Abstract We introduce a Galois connection calculus for language independent 3. Basic GC semantics Basic GCs are primitive abstractions of specification of abstract interpretations used in programming language semantics, properties. Classical examples are the identity abstraction 1[ , formal verification, and static analysis. This Galois connection calculus and its type λ Q . Q S hC system are typed by abstract interpretation. ] , , , , the top abstraction [ , vi hC vi −− − λ − −P − − −. − −P !−!− − hC vi S > hC Categories and Subject Descriptors D.2.4 [Software/Program Verification] λ Q . J General Terms Algorithms, Languages, Reliability, Security, Theory, Verification. , ] , , > , , the join abstraction [C] vi K> hC vi −−−−−λ P . hC vi S J[ Keywords Abstract Interpretation, Galois connection, Static Analysis, Verification. −−−−−!γ} > }(}(C)), }(C), with ↵}(P ) P , γ}(Q) In Abstract , } , , 1. Galois connections in Abstract Interpretation h K ✓i −−! −−↵ h ✓i J K interpretation [3, 4, 6, 7] concrete properties (for example (e.g.) }(Q), the complement abstraction [C] , }(SC), ¬ of computations) are related to abstract properties (e.g. types). The S ¬ h ✓i −−! −¬ }(C), , the finite/infinite sequence abstraction [C] , abstract properties are always sound approximations of the con- h ◆i γ S 1 1 J K crete properties (abstract proofs/static analyzes are always correct }(C1), }(C), with ↵1(P ) , σi σ P i h ✓i ↵ −1 −− h ✓i { | 2 ^ 2 in the concrete) and are sometimes complete (proofs/analyzes of −− − −! J K dom(σ) and γ1(Q) , σ C1 i dom(σ):σi Q , the abstract properties can all be done in the abstract only).
    [Show full text]
  • The Galois Correspondence Between a Ring and Its Spectrum
    THE GALOIS CORRESPONDENCE BETWEEN A RING AND ITS SPECTRUM Let A and X be two sets equipped with a relation R ⊆ A × X. The notation is intended to remind you of the case where A is a commutative ring, X is its spectrum Spec A, and R is given by (f; x) 2 R () f(x) = 0 (i.e., f 2 px): But mathematics is full of other examples, including Galois theory (which is where the phrase \Galois correspondence" in the title comes from). For any subset S ⊆ A, define S0 ⊆ X by S0 := fx 2 X j (f; x) 2 R for all f 2 Sg : Similarly, for any subset Y ⊆ X, define Y 0 ⊆ A by Y 0 := ff 2 A j (f; x) 2 R for all x 2 Y g : In our canonical example, S0 = V (S) (the \zero set" of S), and Y 0 = I(Y ) := T 0 y2Y py. In more intuitive language, Y is the ideal of functions that vanish on Y . These \prime" operations are order reversing: S ⊆ T =) T 0 ⊆ S0; and similarly for subsets of X. Any subset of X of the form S0 will be called closed. (This gives the Zariski topology in our canonical example.) Similarly, any subset of A of the form Y 0 will be called closed. (So the closed subsets of A are the radical ideals in our canonical example.) Note that we always have S ⊆ S00 and Y ⊆ Y 00 for S ⊆ A and Y ⊆ X. I claim that equality holds for closed sets.
    [Show full text]
  • Galois Connections
    1 Galois Connections Roland Backhouse 3rd December, 2002 2 Fusion Many problems are expressed in the form evaluate generate ◦ where generate generates a (possibly infinite) candidate set of solutions, and evaluate selects a best solution. Examples: shortest path ; ◦ (x ) L: 2 ◦ Solution method is to fuse the generation and evaluation processes, eliminating the need to generate all candidate solutions. 3 Conditions for Fusion Fusion is made possible when evaluate is an adjoint in a Galois connection, • generate is expressed as a fixed point. • Solution method typically involves generalising the problem. 4 Definition Suppose = (A; ) and = (B; ) are partially ordered sets and A v B suppose F A B and G B A . Then (F;G) is a Galois connection 2 2 of and iff, for all x B and y A, A B 2 2 F:x y x G:y : v ≡ F is called the lower adjoint. G is the upper adjoint. 5 Examples | Propositional Calculus :p q p :q ≡ p ^ q r q (p r) ) ≡ ( p _ q r (p r) ^ (q r) ) ≡ ) ) p q _ r p ^ :q r ) ≡ ) ) ) ) Examples | Set Theory :S T S :T ⊆ ≡ ⊇ S T U T :S U \ ⊆ ≡ ⊆ [ S T U S U ^ T U [ ⊆ ≡ ⊆ ⊆ S T U S :T U ⊆ [ ≡ \ ⊆ 6 Examples | Number Theory -x y x -y ≤ ≡ ≥ x+y z y z-x ≤ ≡ ≤ x n x n d e ≤ ≡ ≤ x y z x z ^ y z ≤ ≡ ≤ ≤ x y z x z=y for all y > 0 ×" ≤ ≡ ≤ 7 Examples | predicates even:m b (if b then 2 else 1) n m ≡ odd:m b (if b then 1 else 2) n m ( ≡ x S b S if b then fxg else φ 2 ) ≡ ⊇ x S b S if b then U else Unfxg 2 ( ≡ ⊆ S = φ b S if b then φ else U ) ≡ ⊆ S = φ b S if b then U else φ 6 ( ≡ ⊆ ) 8 Examples | programming algebra Languages ("factors") L M N L N=M · ⊆ ≡
    [Show full text]
  • An Algebraic Theory of Complexity for Discrete Optimization Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny
    An Algebraic Theory of Complexity for Discrete Optimization Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny To cite this version: Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny. An Algebraic Theory of Complexity for Discrete Optimization. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2013, vol. 42 (n° 5), pp. 1915-1939. 10.1137/130906398. hal-01122748 HAL Id: hal-01122748 https://hal.archives-ouvertes.fr/hal-01122748 Submitted on 4 Mar 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 12741 To link to this article : DOI :10.1137/130906398 URL : http://dx.doi.org/10.1137/130906398 To cite this version : Cohen, Davis and Cooper, Martin C. and Creed, Paidi and Jeavons, Peter and Zivny, Stanislas An Algebraic Theory of Complexity for Discrete Optimization. (2013) SIAM Journal on Computing, vol. 42 (n° 5).
    [Show full text]
  • Constructive Galois Connections
    ZU064-05-FPR main 23 July 2018 12:36 1 Constructive Galois Connections DAVID DARAIS University of Vermont, USA (e-mail: [email protected]) DAVID VAN HORN University of Maryland, College Park, USA (e-mail: [email protected]) Abstract Galois connections are a foundational tool for structuring abstraction in semantics and their use lies at the heart of the theory of abstract interpretation. Yet, mechanization of Galois connections using proof assistants remains limited to restricted modes of use, preventing their general application in mechanized metatheory and certified programming. This paper presents constructive Galois connections, a variant of Galois connections that is effective both on paper and in proof assistants; is complete with respect toa large subset of classical Galois connections; and enables more general reasoning principles, including the “calculational” style advocated by Cousot. To design constructive Galois connections we identify a restricted mode of use of classical ones which is both general and amenable to mechanization in dependently-typed functional programming languages. Crucial to our metatheory is the addition of monadic structure to Galois connections to control a “specification effect.” Effectful calculations may reason classically, while pure calculations have extractable computational content. Explicitly moving between the worlds of specification and implementation is enabled by our metatheory. To validate our approach, we provide two case studies in mechanizing existing proofs from the literature: the first uses calculational abstract interpretation to design a static analyzer; the second forms a semantic basis for gradual typing. Both mechanized proofs closely follow their original paper-and-pencil counterparts, employ reasoning principles arXiv:1807.08711v1 [cs.PL] 23 Jul 2018 not captured by previous mechanization approaches, support the extraction of verified algorithms, and are novel.
    [Show full text]
  • Lattice Theory
    Chapter 1 Lattice theory 1.1 Partial orders 1.1.1 Binary Relations A binary relation R on a set X is a set of pairs of elements of X. That is, R ⊆ X2. We write xRy as a synonym for (x, y) ∈ R and say that R holds at (x, y). We may also view R as a square matrix of 0’s and 1’s, with rows and columns each indexed by elements of X. Then Rxy = 1 just when xRy. The following attributes of a binary relation R in the left column satisfy the corresponding conditions on the right for all x, y, and z. We abbreviate “xRy and yRz” to “xRyRz”. empty ¬(xRy) reflexive xRx irreflexive ¬(xRx) identity xRy ↔ x = y transitive xRyRz → xRz symmetric xRy → yRx antisymmetric xRyRx → x = y clique xRy For any given X, three of these attributes are each satisfied by exactly one binary relation on X, namely empty, identity, and clique, written respectively ∅, 1X , and KX . As sets of pairs these are respectively the empty set, the set of all pairs (x, x), and the set of all pairs (x, y), for x, y ∈ X. As square X × X matrices these are respectively the matrix of all 0’s, the matrix with 1’s on the leading diagonal and 0’s off the diagonal, and the matrix of all 1’s. Each of these attributes holds of R if and only if it holds of its converse R˘, defined by xRy ↔ yRx˘ . This extends to Boolean combinations of these attributes, those formed using “and,” “or,” and “not,” such as “reflexive and either not transitive or antisymmetric”.
    [Show full text]
  • Galois Connections
    Modern Algebra 2 (MATH 6140) HANDOUT 6 (April 14, 2008) GALOIS CONNECTIONS Let S and T be classes, and let R ⊆ S × T be a relation. For each subset U ⊆ S define U ? = U R := ft 2 T j 8u 2 U((u; t) 2 R)g and for each V ⊆ T define V ? = RV := fs 2 S j 8v 2 V ((s; v) 2 R)g: These are two mappings, ?: P(S) !P(T ) and ?: P(T ) !P(S). It is immediate from the definition of ? that: Lemma 1. U × V ⊆ R () U ⊆ V ? () V ⊆ U ?. The last bi-implication in Lemma 1 motivates the following definition. Definition 2. A Galois connection (GC) between classes S and T is a pair of map- pings ?: P(S) !P(T ) and ?: P(T ) !P(S) such that for all U ⊆ S and V ⊆ T it is the case that U ⊆ V ? () V ⊆ U ? Lemma 3. A Galois connection between S and T arises from a uniquely determined relation in the manner described before Lemma 1. Proof. If a GC between S and T arises from a relation R ⊆ S × T , then, according to Lemma 1, (u; v) 2 R iff fug ⊆ fvg? iff fvg ⊆ fug?. So, given an arbitrary GC, the natural candidate for the associated relation is R = f(u; v) 2 S × T j v 2 fug?g. To see that the GC induced by this relation R is the same as the starting GC we must show that U R = U ? for all U ⊆ S. (Since everything is symmetric so far, the same argument will show RV = V ?.) Note that R is defined so that the desired statement is true for singleton sets: fugR = fug?.
    [Show full text]
  • Order Theory, Galois Connections and Abstract Interpretation
    Order theory Galois Connections GC and Abstract Interpretation Examples Order Theory, Galois Connections and Abstract Interpretation David Henriques Carnegie Mellon University November 7th 2011 fsu-logo David Henriques Order Theory 1/ 40 Order theory Galois Connections GC and Abstract Interpretation Examples Order Theory fsu-logo David Henriques Order Theory 2/ 40 It's not easy, we need a formal treatment of order! Actually, there is an intruder in this list. Can you spot it? Order theory Galois Connections GC and Abstract Interpretation Examples Orders are everywhere 23 I 0 ≤ 1 and 1 ≤ 10 I Two cousins have a common grandfather I 22=7 is a worse approximation of π than 3:141592654 I aardvark comes before zyzzyva I a seraphim ranks above an angel I rock beats scissors I neither f1; 2; 4g or f2; 3; 5g are subsets of one another, but both are subsets of f1; 2; 3; 4; 5g fsu-logo David Henriques Order Theory 3/ 40 It's not easy, we need a formal treatment of order! Order theory Galois Connections GC and Abstract Interpretation Examples Orders are everywhere 23 I 0 ≤ 1 and 1 ≤ 10 I Two cousins have a common grandfather I 22=7 is a worse approximation of π than 3:141592654 I aardvark comes before zyzzyva I a seraphim ranks above an angel I rock beats scissors I neither f1; 2; 4g or f2; 3; 5g are subsets of one another, but both are subsets of f1; 2; 3; 4; 5g Actually, there is an intruder in this list. Can you spot it? fsu-logo David Henriques Order Theory 3/ 40 Order theory Galois Connections GC and Abstract Interpretation Examples Orders are everywhere 23 I 0 ≤ 1 and 1 ≤ 10 I Two cousins have a common grandfather I 22=7 is a worse approximation of π than 3:141592654 I aardvark comes before zyzzyva I a seraphim ranks above an angel I rock beats scissors I neither f1; 2; 4g or f2; 3; 5g are subsets of one another, but both are subsets of f1; 2; 3; 4; 5g Actually, there is an intruder in this list.
    [Show full text]
  • Galois Connections for Phylogenetic Networks and Their Polytopes
    GALOIS CONNECTIONS FOR PHYLOGENETIC NETWORKS AND THEIR POLYTOPES STEFAN FORCEY AND DREW SCALZO Abstract. We describe Galois connections which arise between two kinds of combinatorial structures, both of which generalize trees with labelled leaves, and then apply those connections to a family of polytopes. The graphs we study can be imbued with metric properties or associated to vectors. Famous exam- ples are the Billera-Holmes-Vogtmann metric space of phylogenetic trees, and the Balanced Minimal Evolution polytopes of phylogenetic trees described by Eickmeyer, Huggins, Pachter and Yoshida. Recently the space of trees has been expanded to split networks by Devadoss and Petti, while the def- inition of phylogenetic polytopes has been generalized to encompass 1-nested phylogenetic networks, by Durell and Forcey. The first Galois connection we describe is a reflection between the (unweighted) circular split networks and the 1-nested phylogenetic networks. Another Galois connection exists be- tween certain metric versions of these structures. Reflection between the purely combinatorial posets becomes a coreflection in the geometric case. Our chief contributions here, beyond the discovery of the Galois connections, are: a translation between approaches using PC-trees and networks, a new way to look at weightings on networks, and a fuller characterization of faces of the phylogenetic polytopes. 1. Introduction The simplest structure we consider here is a partition of a finite set into two parts. A tree with labeled leaves represents a collection of such bipartitions, also known as splits, since removing any edge of the tree partitions the leaves via separating the tree into two components. A collection of splits that can be thus displayed by a tree is called pairwise compatible.
    [Show full text]