Algebraic Structure Binary Operation on a Set OPERATIONS
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Arxiv:0704.2561V2 [Math.QA] 27 Apr 2007 Nttto O Xeln Okn Odtos H Eoda Second the Conditions
DUAL FEYNMAN TRANSFORM FOR MODULAR OPERADS J. CHUANG AND A. LAZAREV Abstract. We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich’s dual construction producing graph cohomology classes from a contractible differential graded Frobenius alge- bra. The dual Feynman transform of a modular operad is indeed linear dual to the Feynman transform introduced by Getzler and Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman transform, the dual notion admits an extremely simple presentation via generators and relations; this leads to an explicit and easy description of its algebras. We discuss a further generalization of the dual Feynman transform whose algebras are not neces- sarily contractible. This naturally gives rise to a two-colored graph complex analogous to the Boardman-Vogt topological tree complex. Contents Introduction 1 Notation and conventions 2 1. Main construction 4 2. Twisted modular operads and their algebras 8 3. Algebras over the dual Feynman transform 12 4. Stable graph complexes 13 4.1. Commutative case 14 4.2. Associative case 15 5. Examples 17 6. Moduli spaces of metric graphs 19 6.1. Commutative case 19 6.2. Associative case 21 7. BV-resolution of a modular operad 22 7.1. Basic construction and description of algebras 22 7.2. Stable BV-graph complexes 23 References 26 Introduction arXiv:0704.2561v2 [math.QA] 27 Apr 2007 The relationship between operadic algebras and various moduli spaces goes back to Kontse- vich’s seminal papers [19] and [20] where graph homology was also introduced. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties
A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields). -
Algebraic Structures Lecture 18 Thursday, April 4, 2019 1 Type
Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages Algebraic structures Lecture 18 Thursday, April 4, 2019 In abstract algebra, algebraic structures are defined by a set of elements and operations on those ele- ments that satisfy certain laws. Some of these algebraic structures have interesting and useful computa- tional interpretations. In this lecture we will consider several algebraic structures (monoids, functors, and monads), and consider the computational patterns that these algebraic structures capture. We will look at Haskell, a functional programming language named after Haskell Curry, which provides support for defin- ing and using such algebraic structures. Indeed, monads are central to practical programming in Haskell. First, however, we consider type constructors, and see two new type constructors. 1 Type constructors A type constructor allows us to create new types from existing types. We have already seen several different type constructors, including product types, sum types, reference types, and parametric types. The product type constructor × takes existing types τ1 and τ2 and constructs the product type τ1 × τ2 from them. Similarly, the sum type constructor + takes existing types τ1 and τ2 and constructs the product type τ1 + τ2 from them. We will briefly introduce list types and option types as more examples of type constructors. 1.1 Lists A list type τ list is the type of lists with elements of type τ. We write [] for the empty list, and v1 :: v2 for the list that contains value v1 as the first element, and v2 is the rest of the list. We also provide a way to check whether a list is empty (isempty? e) and to get the head and the tail of a list (head e and tail e). -
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. -
Semilattice Sums of Algebras and Mal'tsev Products of Varieties
Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S. -
Operations Per Se, to See If the Proposed Chi Is Indeed Fruitful
DOCUMENT RESUME ED 204 124 SE 035 176 AUTHOR Weaver, J. F. TTTLE "Addition," "Subtraction" and Mathematical Operations. 7NSTITOTION. Wisconsin Univ., Madison. Research and Development Center for Individualized Schooling. SPONS AGENCY Neional Inst. 1pf Education (ED), Washington. D.C. REPORT NO WIOCIS-PP-79-7 PUB DATE Nov 79 GRANT OS-NIE-G-91-0009 NOTE 98p.: Report from the Mathematics Work Group. Paper prepared for the'Seminar on the Initial Learning of Addition and' Subtraction. Skills (Racine, WI, November 26-29, 1979). Contains occasional light and broken type. Not available in hard copy due to copyright restrictions.. EDPS PRICE MF01 Plum Postage. PC Not Available from EDRS. DEseR.TPT00S *Addition: Behavioral Obiectives: Cognitive Oblectives: Cognitive Processes: *Elementary School Mathematics: Elementary Secondary Education: Learning Theories: Mathematical Concepts: Mathematics Curriculum: *Mathematics Education: *Mathematics Instruction: Number Concepts: *Subtraction: Teaching Methods IDENTIFIERS *Mathematics Education ResearCh: *Number 4 Operations ABSTRACT This-report opens by asking how operations in general, and addition and subtraction in particular, ante characterized for 'elementary school students, and examines the "standard" Instruction of these topics through secondary. schooling. Some common errors and/or "sloppiness" in the typical textbook presentations are noted, and suggestions are made that these probleks could lend to pupil difficulties in understanding mathematics. The ambiguity of interpretation .of number sentences of the VW's "a+b=c". and "a-b=c" leads into a comparison of the familiar use of binary operations with a unary-operator interpretation of such sentences. The second half of this document focuses on points of vier that promote .varying approaches to the development of mathematical skills and abilities within young children. -
Hyperoperations and Nopt Structures
Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles. -
Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions
Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions Section 10. 1, Problems: 1, 2, 3, 4, 10, 11, 29, 36, 37 (fifth edition); Section 11.1, Problems: 1, 2, 5, 6, 12, 13, 31, 40, 41 (sixth edition) The notation ""forOR is bad and misleading. Just think that in the context of boolean functions, the author uses instead of ∨.The integers modulo 2, that is ℤ2 0,1, have an addition where 1 1 0 while 1 ∨ 1 1. AsetA is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S,it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers ℕ,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L. -
N-Algebraic Structures and S-N-Algebraic Structures
N-ALGEBRAIC STRUCTURES AND S-N-ALGEBRAIC STRUCTURES W. B. Vasantha Kandasamy Florentin Smarandache 2005 1 N-ALGEBRAIC STRUCTURES AND S-N-ALGEBRAIC STRUCTURES W. B. Vasantha Kandasamy e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache e-mail: [email protected] 2005 2 CONTENTS Preface 5 Chapter One INTRODUCTORY CONCEPTS 1.1 Group, Smarandache semigroup and its basic properties 7 1.2 Loops, Smarandache Loops and their basic properties 13 1.3 Groupoids and Smarandache Groupoids 23 Chapter Two N-GROUPS AND SMARANDACHE N-GROUPS 2.1 Basic Definition of N-groups and their properties 31 2.2 Smarandache N-groups and some of their properties 50 Chapter Three N-LOOPS AND SMARANDACHE N-LOOPS 3.1 Definition of N-loops and their properties 63 3.2 Smarandache N-loops and their properties 74 Chapter Four N-GROUPOIDS AND SMARANDACHE N-GROUPOIDS 4.1 Introduction to bigroupoids and Smarandache bigroupoids 83 3 4.2 N-groupoids and their properties 90 4.3 Smarandache N-groupoid 99 4.4 Application of N-groupoids and S-N-groupoids 104 Chapter Five MIXED N-ALGEBRAIC STRUCTURES 5.1 N-group semigroup algebraic structure 107 5.2 N-loop-groupoids and their properties 134 5.3 N-group loop semigroup groupoid (glsg) algebraic structures 163 Chapter Six PROBLEMS 185 FURTHER READING 191 INDEX 195 ABOUT THE AUTHORS 209 4 PREFACE In this book, for the first time we introduce the notions of N- groups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure. -
An Introduction to the Theory of Lattice Ý Jinfang Wang £
An Introduction to the Theory of Lattice Ý Jinfang Wang £ Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan May 11, 2006 £ Fax: 81-43-290-3663. E-Mail addresses: [email protected] Ý Partially supported by Grand-in-Aid 15500179 of the Japanese Ministry of Education, Science, Sports and Cul- ture. 1 1 Introduction A lattice1 is a partially ordered set (or poset), in which all nonempty finite subsets have both a supremum (join) and an infimum (meet). Lattices can also be characterized as algebraic structures that satisfy certain identities. Since both views can be used interchangeably, lattice theory can draw upon applications and methods both from order theory and from universal algebra. Lattices constitute one of the most prominent representatives of a series of “lattice-like” structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, and Boolean algebras. 2 Semilattice A semilattice is a partially ordered set within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). Consequently, one speaks of either a join-semilattice or a meet-semilattice. Semilattices may be regarded as a generalization of the more prominent concept of a lattice. In the literature, join-semilattices sometimes are sometimes additionally required to have a least element (the join of the empty set). Dually, meet-semilattices may include a greatest element. 2.1 Definitions Semilattices as posets Ë µ Ü DEFINITION 2.1 (MEET-SEMILATTICE). A poset ´ is a meet-semilattice if for all elements Ë Ü Ý and Ý of , the greatest lower bound (meet) of the set exists. -
'Doing and Undoing' Applied to the Action of Exchange Reveals
mathematics Article How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics John Mason 1,2 1 Department of Mathematics and Statistics, Open University, Milton Keynes MK7 6AA, UK; [email protected] 2 Department of Education, University of Oxford, 15 Norham Gardens, Oxford OX2 6PY, UK Abstract: The theme of ‘undoing a doing’ is applied to the ubiquitous action of exchange, showing how exchange pervades a school’s mathematics curriculum. It is possible that many obstacles encountered in school mathematics arise from an impoverished sense of exchange, for learners and possibly for teachers. The approach is phenomenological, in that the reader is urged to undertake the tasks themselves, so that the pedagogical and mathematical comments, and elaborations, may connect directly to immediate experience. Keywords: doing and undoing; exchange; substitution 1. Introduction Arithmetic is seen here as the study of actions (usually by numbers) on numbers, whereas calculation is an epiphenomenon: useful as a skill, but only to facilitate recognition Citation: Mason, J. How the Theme of relationships between numbers. Particular relationships may, upon articulation and of ‘Doing and Undoing’ Applied to analysis (generalisation), turn out to be properties that hold more generally. This way of the Action of Exchange Reveals thinking sets the scene for the pervasive mathematical theme of inverse, also known as Overlooked Core Ideas in School doing and undoing (Gardiner [1,2]; Mason [3]; SMP [4]). Exploiting this theme brings to Mathematics. Mathematics 2021, 9, the surface core ideas, such as exchange, which, although appearing sporadically in most 1530.