Matrices of Formal Power Series Associated to Binomial Posets
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Research Statement Robert Won
Research Statement Robert Won My research interests are in noncommutative algebra, specifically noncommutative ring the- ory and noncommutative projective algebraic geometry. Throughout, let k denote an alge- braically closed field of characteristic 0. All rings will be k-algebras and all categories and equivalences will be k-linear. 1. Introduction Noncommutative rings arise naturally in many contexts. Given a commutative ring R and a nonabelian group G, the group ring R[G] is a noncommutative ring. The n × n matrices with entries in C, or more generally, the linear transformations of a vector space under composition form a noncommutative ring. Noncommutative rings also arise as differential operators. The ring of differential operators on k[t] generated by multiplication by t and differentiation by t is isomorphic to the first Weyl algebra, A1 := khx; yi=(xy − yx − 1), a celebrated noncommutative ring. From the perspective of physics, the Weyl algebra captures the fact that in quantum mechanics, the position and momentum operators do not commute. As noncommutative rings are a larger class of rings than commutative ones, not all of the same tools and techniques are available in the noncommutative setting. For example, localization is only well-behaved at certain subsets of noncommutative rings called Ore sets. Care must also be taken when working with other ring-theoretic properties. The left ideals of a ring need not be (two-sided) ideals. One must be careful in distinguishing between left ideals and right ideals, as well as left and right noetherianness or artianness. In commutative algebra, there is a rich interplay between ring theory and algebraic geom- etry. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Semicorings and Semicomodules Will Open the Door for Many New Applications in the Future (See [Wor2012] for Recent Applications to Automata Theory)
Semicorings and Semicomodules Jawad Y. Abuhlail∗ Department of Mathematics and Statistics Box # 5046, KFUPM, 31261 Dhahran, KSA [email protected] July 7, 2018 Abstract In this paper, we introduce and investigate semicorings over associative semir- ings and their categories of semicomodules. Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other dif- ficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings. Introduction Coalgebraic structures in general, and categories of comodules for comonads in par- arXiv:1303.3924v1 [math.RA] 15 Mar 2013 ticular, are gaining recently increasing interest [Wis2012]. Although comonads can be defined in arbitrary categories, nice properties are usually obtained in case the comonad under consideration is isomorphic to −• C (or C • −) for some comonoid C [Por2006] in a monoidal category (V, •, I) acting on some category X in a nice way [MW]. However, it can be noticed that the most extensively studied concrete examples are – so far – the categories of coacts (usually called coalgebras) of an endo-functor F on the cate- gory Set of sets (motivated by applications in theoretical computer science [Gum1999] and universal (co)algebra [AP2003]) and categories of comodules for a coring over an associate algebra [BW2003], [Brz2009], i.e. -
On Ring Extensions for Completely Primary Noncommutativerings
ON RING EXTENSIONS FOR COMPLETELY PRIMARY NONCOMMUTATIVERINGS BY E. H. FELLER AND E. W. SWOKOWSKI 0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x~\, x an indeterminate where ax = xa for all aeR,isN primary and then constructing the right quotient ring QCR[x]). The details of these results appear in §§1,7 and 8. The correspond- ing results for the commutative case are given by E. Snapper in [7] and [8]. If Re: A, where A is completely JVprimary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(o) when oeA and ao = era for all aER in the cases where a is algebraic or transcendental over R. These structures are determined in §§ 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, JV or N(R) denotes the union of nilpotent ideals^) of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R. -
Decomposition Spaces, Incidence Algebras and M\" Obius Inversion I
DECOMPOSITION SPACES, INCIDENCE ALGEBRAS AND MOBIUS¨ INVERSION I: BASIC THEORY IMMA GALVEZ-CARRILLO,´ JOACHIM KOCK, AND ANDREW TONKS Abstract. This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and M¨obius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in ∆. Just as the Segal condition expresses composition, the new exactness condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1-simplices, and with coefficients in ∞-groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen S•-construction of an abelian (or stable infinity) category is shown to be a decomposition space. In the second paper in this series we impose further conditions on decomposition spaces, to obtain a general M¨obius inversion principle, and to ensure that the various constructions and results admit a homotopy cardinality. In the third paper we show that the Lawvere–Menni Hopf algebra of M¨obius intervals is the homotopy cardinality of a certain universal decomposition space. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
Combinatorics
09.10.74 Combinatorics Lectures of Mark Dukes Typing and layout by Líney Halla Kristinsdóttir University of Iceland Spring Term 2008 Contents 1 Enumeration1 1.1 Counting Arguments...............................2 1.2 Elementary Counting Coecients........................3 1.3 Compositions & Partitions of an Integer....................7 1.4 The Reection Principle.............................9 1.5 Stirling Numbers................................. 10 1.5.1 Stirling Numbers of the 2nd Kind: S(n; k) ............... 10 1.5.2 Signless Stirling Numbers of the 1st Kind............... 12 1.6 Multinomial Coecients............................. 14 1.7 Generating Functions I.............................. 15 1.8 Generating Functions II............................. 22 2 Permutations & Permutation Statistics 27 2.1 Descents & the Eulerian Polynomial...................... 27 2.2 Cycle structure & left-to-right maxima..................... 32 2.3 Excedances & weak excedances......................... 33 2.4 Inversions & the Major index.......................... 35 2.5 Multisets, Permutations & q-series....................... 39 2.6 Subspaces of a Vector Space.......................... 43 2.7 Permutations as Increasing Binary Trees.................... 45 2.8 Standard Young Tableaux............................ 46 2.9 Robinson-Schensted-Knuth (RSK) correspondence............... 48 i 3 Partially Ordered Sets (Posets) 53 3.1 Posets...................................... 53 3.2 Lattices..................................... 60 3.3 Modular Lattices................................ -
LTCC Enumerative Combinatorics 5 Posets and M¨Obius Inversion
LTCC Enumerative Combinatorics Notes 5 Alex Fink Fall 2015 5 Posets and Mobius¨ inversion Mobius¨ inversion can be viewed as a generalisation of the inclusion-exclusion principle with an apparatus to keep track of how the conditions intersect, as an apparatus to reduce the number of terms. The apparatus, that of partial orders, turns out to be of great combinatorial utility in its own right. 5.1 The inclusion-exclusion principle Often we are in the situation where we have a number of conditions on a set of combinatorial objects, and we have information about the number of objects which satisfy various combinations of these conditions (inclusion), while we want to count the objects satisfying none of the conditions (exclusion), or perhaps satis- fying some but not others. What is known as the sieve method is of general use in this situation: overcount the objects satisfying the conditions, and then make cor- rections and subtract off elements that have been multiply counted, and so forth. The sieve of Eratosthenes gave its name to the class (although, alone, it’s not es- pecially helpful for the enumeration of primes): the primes are the integers which satify none of the conditions of having the forms 2n;3n;5n;7n;::: for n ≥ 2. Let A1;:::;An be subsets of a finite set X. For any non-empty subset J of the index set [n], we put \ AJ = A j; j2J and take A/0 = X by convention. Theorem 5.1 (Inclusion-Exclusion Principle) The number of elements of X ly- ing in none of the sets Ai is equal to jJj ∑ (−1) jAJj: J⊆[n] Proof The expression in the theorem is a linear combination of the cardinalities of the sets AJ, and so we can calculate it by working out, for each x 2 X, the contribution of x to the sum. -
Elementary Proof of a Theorem of Hawkes, Isaacs And¨Ozaydin
Elementary Proof of a Theorem of Hawkes, Isaacs and Ozaydin¨ Matth´evan der Lee Abstract We present an elementary proof of the theorem of Hawkes, Isaacs and Ozaydin,¨ which states that Σ µG(H, K) ≡ 0 mod d, where µG denotes the M¨obius function for the subgroup lattice of a finite group G, H ranges over the conjugates of a given subgroup F of G with [G : F ] divisible by d, and K over the supergroups of the H for which [K : H] divides d. We apply the theorem to obtain a result on the number of solutions of |hH,gi| | n, for said H and a natural number n. The present version of the article includes an additional result on a quantity studied by K.S. Brown. Keywords: M¨obius function, arithmetic functions, subgroup lattice 2010 MSC: 05E15, 11A25, 20D30 1 Introduction The purpose of this note is to present a simple proof for the theorem of Hawkes, Isaacs and Ozaydin,¨ an important tool in the area of counting problems in finite groups. We refer to [3], where the result has appeared as Theorem 5.1, for background information regarding the subject. The exposition uses minimal group theory. The ingredients for the proof are Burnside’s Lemma, the incidence algebra of a finite partially ordered set (of which we use only the basic properties), and a feature of arithmetic functions: Corollary 3.4. Next, we derive a result on the number of solutions g of |hH,gi| | n, for a finite group G, a subgroup H ≤ G, and n | |G| (Theorem 6.1), and some results on a quantity studied by K.S. -
The Incidence Algebra and the Möbius Function
Monday 2/25 The Incidence Algebra Many enumerative properties of posets P can be expressed in terms of a ring called its incidence algebra. Definition 1. Let P be a locally finite poset and let Int(P ) denote the set of intervals of P . The incidence algebra I(P ) is the set of functions f : Int(P ) ! C. I'll abbreviate f([x; y]) by f(x; y). (For convenience, we set f(x; y) = 0 if x 6≤ y.) This is a C-vector space with pointwise addition, subtraction and scalar multiplication. It can be made into an associative algebra by the convolution product: (f ∗ g)(x; y) = f(x; z)g(z; y): z2X[x;y] Proposition 1. Convolution is associative. Proof. [(f ∗ g) ∗ h](x; y) = (f ∗ g)(x; z) · h(z; y) z2X[x;y] = f(x; w)g(w; z) h(z; y) 0 1 z2X[x;y] w2X[x;z] @ A = f(x; w) g(w; z)h(z; y) 0 1 w2X[x;y] z2X[w;y] = f(x; w) @· (g ∗ h)(w; y) A w2X[x;y] = [f ∗ (g ∗ h)](x; y): Proposition 2. f 2 I(P ) is invertible if and only if f(x; x) 6= 0 for all x. Proof. If f is invertible with inverse g, then (f ∗g)(x; x) = f(x; x)g(x; x) for all x, so in particular f(x; x) 6= 0. OTOH, if f(x; x) 6= 0, then we can define a left inverse g inductively: g(x; x) = 1=f(x; x), and for x < y, we want to have (g ∗ f)(x; y) = 0 = g(x; z)f(z; y) x z y ≤X≤ = g(x; y)f(x; x) + f(x; z)g(z; y) x z<y ≤X so define 1 g(x; y) = − g(x; z)f(z; y): f(x; x) x≤zy X The identity element of I(P ) is the Kronecker delta function: 1 if x = y; δ(x; y) = (0 if x 6= y: The zeta function is defined as 1 if x ≤ y; ζ(x; y) = (0 if x 6≤ y and the eta function is 1 if x < y; η(x; y) = (0 if x 6< y; i.e., η = ζ − δ. -
The Theory of Compositionals
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 122 (1993) 59-87 59 North-Holland The theory of compositionals William Y.C. Chen C-3. Mail Stop 8265, Los Alanm Nationul Lahorutor_v, Los Alumos, NM 87545. USA Received 13 December 1991 Abstract By introducing the notion of compositionals we obtain a combinatorial interpretation of plethysm of formal power series in infinitely many variables. The following related problems are studied: P6lya’s theorem on the plethysm of cycle indices, plethystic inverse, the inverse of a sequence of delta series, the plethystic analog of the partition lattice, the reduced incidence algebra of the plethystic lattice, and the plethystic Hopf algebra. We also introduce plethystic trees, enriched plethystic trees and plethystic Schriider trees. An involution for plethystic Schroder trees is devised, which leads to a combinatorial expansion for the plethystic inverse of a series _j’(.xl, x2, .) containing the factor x1. 1. Introduction Formal power series in infinitely many variables arise naturally in enumerative combinatorics, especially in enumeration problems related to the symmetric group. The composition or plethysm of two series in many variables is defined as follows. Let f(xr ) x2, . .) and g(x1 ,x2, . ) be the two series such that g(xr , x2, . .) has no constant term, then the plethysm of ,f and g is given by fCd(xI>xz> ... )=f‘(g,(x,,.~,,...),g,(x,,x,,...), . ..). where gk(x1x2, ...)=g(xk,x2k,x3k, . ..). A series without constant term will be called a delta series. For simplicity, we shall writef(g(xr,x,, .