Disjoint-Union Partial Algebras
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Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of Abbreviations
Notes: c F.P. Greenleaf, 2000-2014 v43-s14sets.tex (version 1/1/2014) Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of abbreviations. Below we list some standard math symbols that will be used as shorthand abbreviations throughout this course. means “for all; for every” • ∀ means “there exists (at least one)” • ∃ ! means “there exists exactly one” • ∃ s.t. means “such that” • = means “implies” • ⇒ means “if and only if” • ⇐⇒ x A means “the point x belongs to a set A;” x / A means “x is not in A” • ∈ ∈ N denotes the set of natural numbers (counting numbers) 1, 2, 3, • · · · Z denotes the set of all integers (positive, negative or zero) • Q denotes the set of rational numbers • R denotes the set of real numbers • C denotes the set of complex numbers • x A : P (x) If A is a set, this denotes the subset of elements x in A such that •statement { ∈ P (x)} is true. As examples of the last notation for specifying subsets: x R : x2 +1 2 = ( , 1] [1, ) { ∈ ≥ } −∞ − ∪ ∞ x R : x2 +1=0 = { ∈ } ∅ z C : z2 +1=0 = +i, i where i = √ 1 { ∈ } { − } − 1.2 Basic facts from set theory. Next we review the basic definitions and notations of set theory, which will be used throughout our discussions of algebra. denotes the empty set, the set with nothing in it • ∅ x A means that the point x belongs to a set A, or that x is an element of A. • ∈ A B means A is a subset of B – i.e. -
Quantum Heaps, Cops and Heapy Categories
Mathematical Communications 12(2007), 1-9 1 Quantum heaps, cops and heapy categories Zoran Skodaˇ ∗ Abstract. A heap is a structure with a ternary operation which is intuitively a group with a forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in a monoidal category which is in the case of vector spaces to a quantum heap about what a coalge- bra is to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly, the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category. Key words: quantum algebra, rings, algebras AMS subject classifications: 20N10, 16A24 Received January 25, 2007 Accepted February 1, 2007 1. Classical background: heaps A reference for this section is [1]. 1.1 Before forgetting: group as heap. Let G be a group. Then the ternary operation t : G × G × G → G given by t(a, b, c)=ab−1c, (1) satisfies the following relations: t(b, b, c)=c = t(c, b, b) (2) t(a, b, t(c, d, e)) = t(t(a, b, c),d,e) Aheap(H, t) is a pair of nonempty set H and a ternary operation t : H × H × H satisfying relation (2). A morphism f :(H, t) → (H,t) of heaps is a set map f : H → H satisfying t ◦ (f × f × f)=f ◦ t. -
MATH 361 Homework 9
MATH 361 Homework 9 Royden 3.3.9 First, we show that for any subset E of the real numbers, Ec + y = (E + y)c (translating the complement is equivalent to the complement of the translated set). Without loss of generality, assume E can be written as c an open interval (e1; e2), so that E + y is represented by the set fxjx 2 (−∞; e1 + y) [ (e2 + y; +1)g. This c is equal to the set fxjx2 = (e1 + y; e2 + y)g, which is equivalent to the set (E + y) . Second, Let B = A − y. From Homework 8, we know that outer measure is invariant under translations. Using this along with the fact that E is measurable: m∗(A) = m∗(B) = m∗(B \ E) + m∗(B \ Ec) = m∗((B \ E) + y) + m∗((B \ Ec) + y) = m∗(((A − y) \ E) + y) + m∗(((A − y) \ Ec) + y) = m∗(A \ (E + y)) + m∗(A \ (Ec + y)) = m∗(A \ (E + y)) + m∗(A \ (E + y)c) The last line follows from Ec + y = (E + y)c. Royden 3.3.10 First, since E1;E2 2 M and M is a σ-algebra, E1 [ E2;E1 \ E2 2 M. By the measurability of E1 and E2: ∗ ∗ ∗ c m (E1) = m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c m (E2) = m (E2 \ E1) + m (E2 \ E1) ∗ ∗ ∗ ∗ c ∗ c m (E1) + m (E2) = 2m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c ∗ c = m (E1 \ E2) + [m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2)] c c Second, E1 \ E2, E1 \ E2, and E1 \ E2 are disjoint sets whose union is equal to E1 [ E2. -
Ordered Groupoids and the Holomorph of an Inverse Semigroup
ORDERED GROUPOIDS AND THE HOLOMORPH OF AN INVERSE SEMIGROUP N.D. GILBERT AND E.A. MCDOUGALL ABSTRACT. We present a construction for the holomorph of an inverse semi- group, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that pre- serve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids. INTRODUCTION The holomorph of a group G is the semidirect product of Hol(G) = Aut(G) n G of G and its automorphism group (with the natural action of Aut(G) on G). The embedding of Aut(G) into the symmetric group ΣG on G extends to an embedding of Hol(G) into ΣG where g 2 G is identified with its (right) Cayley representation ρg : a 7! ag. Then Hol(G) is the normalizer of G in ΣG ([15, Theorem 9.17]). A second interesting characterization of Hol(G) is due to Baer [1] (see also [3] and [15, Exercise 520]). The heap operation on G is the ternary operation defined by ha; b; ci = ab−1c. Baer shows that a subset of G is closed under the heap operation if and only if it is a coset of some subgroup, and that the subset of ΣG that preserves h· · ·i is precisely Hol(G): that is, if σ 2 ΣG, then for all a; b; c; 2 G we have ha; b; ciσ = haσ; bσ; cσi if and only if σ 2 Hol(G). -
Solutions to Homework 2 Math 55B
Solutions to Homework 2 Math 55b 1. Give an example of a subset A ⊂ R such that, by repeatedly taking closures and complements. Take A := (−5; −4)−Q [[−4; −3][f2g[[−1; 0][(1; 2)[(2; 3)[ (4; 5)\Q . (The number of intervals in this sets is unnecessarily large, but, as Kevin suggested, taking a set that is complement-symmetric about 0 cuts down the verification in half. Note that this set is the union of f0g[(1; 2)[(2; 3)[ ([4; 5] \ Q) and the complement of the reflection of this set about the the origin). Because of this symmetry, we only need to verify that the closure- complement-closure sequence of the set f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q consists of seven distinct sets. Here are the (first) seven members of this sequence; from the eighth member the sets start repeating period- ically: f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q ; f0g [ [1; 3] [ [4; 5]; (−∞; 0) [ (0; 1) [ (3; 4) [ (5; 1); (−∞; 1] [ [3; 4] [ [5; 1); (1; 3) [ (4; 5); [1; 3] [ [4; 5]; (−∞; 1) [ (3; 4) [ (5; 1). Thus, by symmetry, both the closure- complement-closure and complement-closure-complement sequences of A consist of seven distinct sets, and hence there is a total of 14 different sets obtained from A by taking closures and complements. Remark. That 14 is the maximal possible number of sets obtainable for any metric space is the Kuratowski complement-closure problem. This is proved by noting that, denoting respectively the closure and com- plement operators by a and b and by e the identity operator, the relations a2 = a; b2 = e, and aba = abababa take place, and then one can simply list all 14 elements of the monoid ha; b j a2 = a; b2 = e; aba = abababai. -
ON the CONSTRUCTION of NEW TOPOLOGICAL SPACES from EXISTING ONES Consider a Function F
ON THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM EXISTING ONES EMILY RIEHL Abstract. In this note, we introduce a guiding principle to define topologies for a wide variety of spaces built from existing topological spaces. The topolo- gies so-constructed will have a universal property taking one of two forms. If the topology is the coarsest so that a certain condition holds, we will give an elementary characterization of all continuous functions taking values in this new space. Alternatively, if the topology is the finest so that a certain condi- tion holds, we will characterize all continuous functions whose domain is the new space. Consider a function f : X ! Y between a pair of sets. If Y is a topological space, we could define a topology on X by asking that it is the coarsest topology so that f is continuous. (The finest topology making f continuous is the discrete topology.) Explicitly, a subbasis of open sets of X is given by the preimages of open sets of Y . With this definition, a function W ! X, where W is some other space, is continuous if and only if the composite function W ! Y is continuous. On the other hand, if X is assumed to be a topological space, we could define a topology on Y by asking that it is the finest topology so that f is continuous. (The coarsest topology making f continuous is the indiscrete topology.) Explicitly, a subset of Y is open if and only if its preimage in X is open. With this definition, a function Y ! Z, where Z is some other space, is continuous if and only if the composite function X ! Z is continuous. -
MAD FAMILIES CONSTRUCTED from PERFECT ALMOST DISJOINT FAMILIES §1. Introduction. a Family a of Infinite Subsets of Ω Is Called
The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 MAD FAMILIES CONSTRUCTED FROM PERFECT ALMOST DISJOINT FAMILIES JORG¨ BRENDLE AND YURII KHOMSKII 1 Abstract. We prove the consistency of b > @1 together with the existence of a Π1- definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Ques- tion 16]. For the proof we construct a mad family in L which is an @1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number aB , defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of aB < b (and hence, aB < a). x1. Introduction. A family A of infinite subsets of ! is called almost disjoint (a.d.) if any two elements a; b of A have finite intersection. A family A is called maximal almost disjoint, or mad, if it is an infinite a.d. family which is maximal with respect to that property|in other words, 8a 9b 2 A (ja \ bj = !). The starting point of this paper is the following theorem of Adrian Mathias [11, Corollary 4.7]: Theorem 1.1 (Mathias). There are no analytic mad families. 1 On the other hand, it is easy to see that in L there is a Σ2 definable mad family. In [12, Theorem 8.23], Arnold Miller used a sophisticated method to 1 prove the seemingly stronger result that in L there is a Π1 definable mad family. -
Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations on R with All Classes Countable Is the Main Result of the Paper
CARDINALITY OF WELLORDERED DISJOINT UNIONS OF QUOTIENTS OF SMOOTH EQUIVALENCE RELATIONS WILLIAM CHAN AND STEPHEN JACKSON Abstract. Assume ZF + AD+ + V = L(P(R)). Let ≈ denote the relation of being in bijection. Let κ ∈ ON and hEα : α<κi be a sequence of equivalence relations on R with all classes countable and for all α<κ, R R R R R /Eα ≈ . Then the disjoint union Fα<κ /Eα is in bijection with × κ and Fα<κ /Eα has the J´onsson property. + <ω Assume ZF + AD + V = L(P(R)). A set X ⊆ [ω1] 1 has a sequence hEα : α<ω1i of equivalence relations on R such that R/Eα ≈ R and X ≈ F R/Eα if and only if R ⊔ ω1 injects into X. α<ω1 ω ω Assume AD. Suppose R ⊆ [ω1] × R is a relation such that for all f ∈ [ω1] , Rf = {x ∈ R : R(f,x)} ω is nonempty and countable. Then there is an uncountable X ⊆ ω1 and function Φ : [X] → R which uniformizes R on [X]ω: that is, for all f ∈ [X]ω, R(f, Φ(f)). Under AD, if κ is an ordinal and hEα : α<κi is a sequence of equivalence relations on R with all classes ω R countable, then [ω1] does not inject into Fα<κ /Eα. 1. Introduction The original motivation for this work comes from the study of a simple combinatorial property of sets using only definable methods. The combinatorial property of concern is the J´onsson property: Let X be any n n <ω n set. -
Disjoint Union / Find Equivalence Relations Equivalence Classes
Equivalence Relations Disjoint Union / Find • A relation R is defined on set S if for CSE 326 every pair of elements a, b∈S, a R b is Data Structures either true or false. Unit 13 • An equivalence relation is a relation R that satisfies the 3 properties: › Reflexive: a R a for all a∈S Reading: Chapter 8 › Symmetric: a R b iff b R a; for all a,b∈S › Transitive: a R b and b R c implies a R c 2 Dynamic Equivalence Equivalence Classes Problem • Given an equivalence relation R, decide • Starting with each element in a singleton set, whether a pair of elements a,b∈S is and an equivalence relation, build the equivalence classes such that a R b. • Requires two operations: • The equivalence class of an element a › Find the equivalence class (set) of a given is the subset of S of all elements element › Union of two sets related to a. • It is a dynamic (on-line) problem because the • Equivalence classes are disjoint sets sets change during the operations and Find must be able to cope! 3 4 Disjoint Union - Find Union • Maintain a set of disjoint sets. • Union(x,y) – take the union of two sets › {3,5,7} , {4,2,8}, {9}, {1,6} named x and y • Each set has a unique name, one of its › {3,5,7} , {4,2,8}, {9}, {1,6} members › Union(5,1) › {3,5,7} , {4,2,8}, {9}, {1,6} {3,5,7,1,6}, {4,2,8}, {9}, 5 6 Find An Application • Find(x) – return the name of the set • Build a random maze by erasing edges. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
Handout #4: Connected Metric Spaces
Connected Sets Math 331, Handout #4 You probably have some intuitive idea of what it means for a metric space to be \connected." For example, the real number line, R, seems to be connected, but if you remove a point from it, it becomes \disconnected." However, it is not really clear how to define connected metric spaces in general. Furthermore, we want to say what it means for a subset of a metric space to be connected, and that will require us to look more closely at subsets of metric spaces than we have so far. We start with a definition of connected metric space. Note that, similarly to compactness and continuity, connectedness is actually a topological property rather than a metric property, since it can be defined entirely in terms of open sets. Definition 1. Let (M; d) be a metric space. We say that (M; d) is a connected metric space if and only if M cannot be written as a disjoint union N = X [ Y where X and Y are both non-empty open subsets of M. (\Disjoint union" means that M = X [ Y and X \ Y = ?.) A metric space that is not connected is said to be disconnnected. Theorem 1. A metric space (M; d) is connected if and only if the only subsets of M that are both open and closed are M and ?. Equivalently, (M; d) is disconnected if and only if it has a non-empty, proper subset that is both open and closed. Proof. Suppose (M; d) is a connected metric space. -
Real Analysis 1 Math 623 Notes by Patrick Lei, May 2020
Real Analysis 1 Math 623 Notes by Patrick Lei, May 2020 Lectures by Robin Young, Fall 2018 University of Massachusetts, Amherst Disclaimer These notes are a May 2020 transcription of handwritten notes taken during lecture in Fall 2018. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 Lebesgue Measure • 3 1.1 Motivation for the Course • 3 1.2 Basic Notions • 4 1.3 Rectangles • 5 1.4 Outer Measure • 6 1.5 Measurable Sets • 7 1.6 Measurable Functions • 10 2 Integration • 13 2.1 Defining the Integral • 13 2.2 Some Convergence Results • 15 2.3 Vector Space of Integrable Functions • 16 2.4 Symmetries • 18 3 Differentiation • 20 3.1 Integration and Differentiation • 20 3.2 Kernels • 22 3.3 Differentiation • 26 3.4 Bounded Variation • 28 3.5 Absolute Continuity • 30 4 General Measures • 33 4.1 Outer Measures • 34 4.2 Results and Applications • 35 4.3 Signed Measures • 36 2 1 Lebesgue Measure 1.1 Motivation for the Course We will consider some examples that show us some fundamental problems in analysis. Example 1.1 (Fourier Series). Let f be periodic on [0, 2π]. Then if f is sufficiently nice, we can write inx f(x) = ane , n X 1 2π inx where an = 2π 0 f(x)e dx.