Totally Ordered Commutative Monoids
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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. -
Section 8.6 What Is a Partial Order?
Announcements ICS 6B } Regrades for Quiz #3 and Homeworks #4 & Boolean Algebra & Logic 5 are due Today Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 8 – Ch. 8.6, 11.6 Lecture Set 8 - Chpts 8.6, 11.1 2 Today’s Lecture } Chapter 8 8.6, Chapter 11 11.1 ● Partial Orderings 8.6 Chapter 8: Section 8.6 ● Boolean Functions 11.1 Partial Orderings (Continued) Lecture Set 8 - Chpts 8.6, 11.1 3 What is a Partial Order? Some more defintions Let R be a relation on A. The R is a partial order iff R is: } If A,R is a poset and a,b are A, reflexive, antisymmetric, & transitive we say that: } A,R is called a partially ordered set or “poset” ● “a and b are comparable” if ab or ba } Notation: ◘ i.e. if a,bR and b,aR ● If A, R is a poset and a and b are 2 elements of A ● “a and b are incomparable” if neither ab nor ba such that a,bR, we write a b instead of aRb ◘ i.e if a,bR and b,aR } If two objects are always related in a poset it is called a total order, linear order or simple order. NOTE: it is not required that two things be related under a partial order. ● In this case A,R is called a chain. ● i.e if any two elements of A are comparable ● That’s the “partial” of it. ● So for all a,b A, it is true that a,bR or b,aR 5 Lecture Set 8 - Chpts 8.6, 11.1 6 1 Now onto more examples… More Examples Let Aa,b,c,d and let R be the relation Let A0,1,2,3 and on A represented by the diagraph Let R0,01,1, 2,0,2,2,2,33,3 The R is reflexive, but We draw the associated digraph: a b not antisymmetric a,c &c,a It is easy to check that R is 0 and not 1 c d transitive d,cc,a, but not d,a Refl. -
[Math.NT] 1 Nov 2006
ADJOINING IDENTITIES AND ZEROS TO SEMIGROUPS MELVYN B. NATHANSON Abstract. This note shows how iteration of the standard process of adjoining identities and zeros to semigroups gives rise naturally to the lexicographical ordering on the additive semigroups of n-tuples of nonnegative integers and n-tuples of integers. 1. Semigroups with identities and zeros A binary operation ∗ on a set S is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a,b,c ∈ S. A semigroup is a nonempty set with an associative binary operation ∗. The semigroup is abelian if a ∗ b = b ∗ a for all a,b ∈ S. The trivial semigroup S0 consists of a single element s0 such that s0 ∗ s0 = s0. Theorems about abstract semigroups are, in a sense, theorems about the pure process of multiplication. An element u in a semigroup S is an identity if u ∗ a = a ∗ u = a for all a ∈ S. If u and u′ are identities in a semigroup, then u = u ∗ u′ = u′ and so a semigroup contains at most one identity. A semigroup with an identity is called a monoid. If S is a semigroup that is not a monoid, that is, if S does not contain an identity element, there is a simple process to adjoin an identity to S. Let u be an element not in S and let I(S,u)= S ∪{u}. We extend the binary operation ∗ from S to I(S,u) by defining u ∗ a = a ∗ u = a for all a ∈ S, and u ∗ u = u. Then I(S,u) is a monoid with identity u. -
A Unified Approach to Various Generalizations of Armendariz Rings
Bull. Aust. Math. Soc. 81 (2010), 361–397 doi:10.1017/S0004972709001178 A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS GREG MARKS ˛, RYSZARD MAZUREK and MICHAŁ ZIEMBOWSKI (Received 5 February 2009) Abstract Let R be a ring, S a strictly ordered monoid, and ! V S ! End.R/ a monoid homomorphism. The skew generalized power series ring RTTS;!UU is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the .S; !/-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of .S; !/- Armendariz rings and obtain various necessary or sufficient conditions for a ring to be .S; !/-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal. 2000 Mathematics subject classification: primary 16S99, 16W60; secondary 06F05, 16P60, 16S36, 16U80, 20M25. Keywords and phrases: skew generalized power series ring, .S; !/-Armendariz, semicommutative, 2-primal, reversible, reduced, uniserial. 1. Introduction In 1974 Armendariz noted in [3] that whenever the product of two polynomials over a reduced ring R (that is, a ring without nonzero nilpotent elements) is zero, then the products of their coefficients are all zero, that is, in the polynomial ring RTxU the following holds: for any f .x/ D P a xi ; g.x/ D P b x j 2 RTxU; i j (∗) if f .x/g.x/ D 0; then ai b j D 0 for all i; j: Nowadays the property (∗) is known as the Armendariz condition, and rings R that satisfy (∗) are called Armendariz rings. -
Discriminants and the Monoid of Quadratic Rings
DISCRIMINANTS AND THE MONOID OF QUADRATIC RINGS JOHN VOIGHT Abstract. We consider the natural monoid structure on the set of quadratic rings over an arbitrary base scheme and characterize this monoid in terms of discriminants. Quadratic field extensions K of Q are characterized by their discriminants. Indeed, there is a bijection 8Separable quadratic9 < = ∼ × ×2 algebras over Q −! Q =Q : up to isomorphism ; p 2 ×2 Q[ d] = Q[x]=(x − d) 7! dQ wherep a separable quadratic algebra over Q is either a quadratic field extension or the algebra Q[ 1] ' Q × Q of discriminant 1. In particular, the set of isomorphism classes of separable quadratic extensions of Q can be given the structure of an elementary abelian 2-group, with identity element the class of Q × Q: we have simply p p p Q[ d1] ∗ Q[ d2] = Q[ d1d2] p × ×2 up to isomorphism. If d1; d2; dp1d2 2pQ n Q then Q( d1d2) sits as the third quadratic subfield of the compositum Q( d1; d2): p p Q( d1; d2) p p p Q( d1) Q( d1d2) Q( d2) Q p Indeed, ifpσ1 isp the nontrivial elementp of Gal(Q( d1)=Q), then therep is a unique extension of σ1 to Q( d1; d2) leaving Q( d2) fixed, similarly with σ2, and Q( d1d2) is the fixed field of the composition σ1σ2 = σ2σ1. This characterization of quadratic extensions works over any base field F with char F 6= 2 and is summarized concisely in the Kummer theory isomorphism H1(Gal(F =F ); {±1g) = Hom(Gal(F =F ); {±1g) ' F ×=F ×2: On the other hand, over a field F with char F = 2, all separable quadratic extensions have trivial discriminant and instead they are classified by the (additive) Artin-Schreier group F=}(F ) where }(F ) = fr + r2 : r 2 F g Date: January 17, 2021. -
Encyclopaedia of Mathematical Sciences Volume 135 Invariant
Encyclopaedia of Mathematical Sciences Volume 135 Invariant Theory and Algebraic Transformation Groups VI Subseries Editors: R.V. Gamkrelidze V.L. Popov Martin Lorenz Multiplicative Invariant Theory 123 Author Martin Lorenz Department of Mathematics Temple Universit y Philadelphia, PA 19122, USA e-mail: [email protected] Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze Mathematics Subject Classification (2000): Primary: 13A50 Secondary: 13H10, 13D45, 20C10, 12F20 ISSN 0938-0396 ISBN 3-540-24323-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media GmbH springeronline.com ©Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro- tective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover Design: E. Kirchner, Heidelberg, Germany Printed on acid-free paper 46/3142 YL 5 4 3 2 1 0 To my mother, Martha Lorenz, and to the memory of my father, Adolf Lorenz (1925 – 2001) Preface Multiplicative invariant theory, as a research area in its own right, is of relatively recent vintage: the systematic investigation of multiplicative invariants was initiated by Daniel Farkas in the 1980s. -
Symmetric Ideals and Numerical Primary Decomposition
SYMMETRIC IDEALS AND NUMERICAL PRIMARY DECOMPOSITION A Thesis Presented to The Academic Faculty by Robert C. Krone In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mathematics Georgia Institute of Technology August 2015 Copyright c 2015 by Robert C. Krone SYMMETRIC IDEALS AND NUMERICAL PRIMARY DECOMPOSITION Approved by: Professor Anton Leykin, Advisor Professor Stavros Garoufalidis School of Mathematics School of Mathematics Georgia Institute of Technology Georgia Institute of Technology Professor Josephine Yu Professor Santosh Vempala School of Mathematics College of Computing Georgia Institute of Technology Georgia Institute of Technology Professor Greg Blekherman Date Approved: 26 May 2015 School of Mathematics Georgia Institute of Technology ACKNOWLEDGEMENTS There are many people who deserve recognition for their parts in my completion of the Ph.D. thesis. First I thank my advisor Anton Leykin for his guidance and support, for supplying interesting and fruitful research problems, for pushing me to go to conferences, give talks and meet potential collaborators, and for putting up with my procrastination and habitual tardiness. I would also like to acknowledge the other research collaborators who contributed to the work that appears in this thesis. These are Jan Draisma, Rob Eggermont, Jon Hauenstein, Chris Hillar and Thomas Kahle. Thanks to Jan Draisma for giving the me the opportunity to work with him and his group at TU Eindhoven during the spring of 2013. Thanks to my thesis committee members Greg Blekherman, Stavros Garoufalidis, Anton Leykin, Santosh Vempala and Josephine Yu, for taking the time to be a part of this process. Gratitude goes to all of my friends at Georgia Tech who worked very hard to make my graduate school experience interesting, both mathematically and otherwise. -
Arxiv:1811.03543V1 [Math.LO]
PREDICATIVE WELL-ORDERING NIK WEAVER Abstract. Confusion over the predicativist conception of well-ordering per- vades the literature and is responsible for widespread fundamental miscon- ceptions about the nature of predicative reasoning. This short note aims to explain the core fallacy, first noted in [9], and some of its consequences. 1. Predicativism Predicativism arose in the early 20th century as a response to the foundational crisis which resulted from the discovery of the classical paradoxes of naive set theory. It was initially developed in the writings of Poincar´e, Russell, and Weyl. Their central concern had to do with the avoidance of definitions they considered to be circular. Most importantly, they forbade any definition of a real number which involves quantification over all real numbers. This version of predicativism is sometimes called “predicativism given the nat- ural numbers” because there is no similar prohibition against defining a natural number by means of a condition which quantifies over all natural numbers. That is, one accepts N as being “already there” in some sense which is sufficient to void any danger of vicious circularity. In effect, predicativists of this type consider un- countable collections to be proper classes. On the other hand, they regard countable sets and constructions as unproblematic. 2. Second order arithmetic Second order arithmetic, in which one has distinct types of variables for natural numbers (a, b, ...) and for sets of natural numbers (A, B, ...), is thus a good setting for predicative reasoning — predicative given the natural numbers, but I will not keep repeating this. Here it becomes easier to frame the restriction mentioned above in terms of P(N), the power set of N, rather than in terms of R. -
Learning Binary Relations and Total Orders
Learning Binary Relations and Total Orders Sally A Goldman Ronald L Rivest Department of Computer Science MIT Lab oratory for Computer Science Washington University Cambridge MA St Louis MO rivesttheorylcsmitedu sgcswustledu Rob ert E Schapire ATT Bell Lab oratories Murray Hill NJ schapireresearchattcom Abstract We study the problem of learning a binary relation b etween two sets of ob jects or b etween a set and itself We represent a binary relation b etween a set of size n and a set of size m as an n m matrix of bits whose i j entry is if and only if the relation holds b etween the corresp onding elements of the twosetsWe present p olynomial prediction algorithms for learning binary relations in an extended online learning mo del where the examples are drawn by the learner by a helpful teacher by an adversary or according to a uniform probability distribution on the instance space In the rst part of this pap er we present results for the case that the matrix of the relation has at most k rowtyp es We present upp er and lower b ounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended online learning mo del Furthermore we describ e a technique that simplies the pro of of exp ected mistake b ounds against a randomly chosen query sequence In the second part of this pap er we consider the problem of learning a binary re lation that is a total order on a set We describ e a general technique using a fully p olynomial randomized approximation scheme fpras to implement a randomized -
Order Relations and Functions
Order Relations and Functions Problem Session Tonight 7:00PM – 7:50PM 380-380X Optional, but highly recommended! Recap from Last Time Relations ● A binary relation is a property that describes whether two objects are related in some way. ● Examples: ● Less-than: x < y ● Divisibility: x divides y evenly ● Friendship: x is a friend of y ● Tastiness: x is tastier than y ● Given binary relation R, we write aRb iff a is related to b by relation R. Order Relations “x is larger than y” “x is tastier than y” “x is faster than y” “x is a subset of y” “x divides y” “x is a part of y” Informally An order relation is a relation that ranks elements against one another. Do not use this definition in proofs! It's just an intuition! Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 1 ≤ 5 and 5 ≤ 8 Properties of Order Relations x ≤ y 1 ≤ 5 and 5 ≤ 8 1 ≤ 8 Properties of Order Relations x ≤ y 42 ≤ 99 and 99 ≤ 137 Properties of Order Relations x ≤ y 42 ≤ 99 and 99 ≤ 137 42 ≤ 137 Properties of Order Relations x ≤ y x ≤ y and y ≤ z Properties of Order Relations x ≤ y x ≤ y and y ≤ z x ≤ z Properties of Order Relations x ≤ y x ≤ y and y ≤ z x ≤ z Transitivity Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 1 ≤ 1 Properties of Order Relations x ≤ y 42 ≤ 42 Properties of Order Relations x ≤ y 137 ≤ 137 Properties of Order Relations x ≤ y x ≤ x Properties of Order Relations x ≤ y x ≤ x Reflexivity Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 19 ≤ 21 Properties of Order Relations x ≤ y 19 ≤ 21 21 ≤ 19? Properties of Order Relations x ≤ y 19 ≤ 21 21 ≤ 19? Properties of Order Relations x ≤ y 42 ≤ 137 Properties of Order Relations x ≤ y 42 ≤ 137 137 ≤ 42? Properties of Order Relations x ≤ y 42 ≤ 137 137 ≤ 42? Properties of Order Relations x ≤ y 137 ≤ 137 Properties of Order Relations x ≤ y 137 ≤ 137 137 ≤ 137? Properties of Order Relations x ≤ y 137 ≤ 137 137 ≤ 137 Antisymmetry A binary relation R over a set A is called antisymmetric iff For any x ∈ A and y ∈ A, If xRy and y ≠ x, then yRx. -
On the Hall Algebra of Semigroup Representations Over F1
ON THE HALL ALGEBRA OF SEMIGROUP REPRESENTATIONS OVER F1 MATT SZCZESNY Abstract. Let A be a finitely generated semigroup with 0. An A–module over F1 (also called an A–set), is a pointed set (M, ∗) together with an action of A. We define and study the Hall algebra HA of the category CA of finite A–modules. HA is shown to be the universal enveloping algebra of a Lie algebra nA, called the Hall Lie algebra of CA. In the case of the hti - the free monoid on one generator hti, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent hti-modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when A is a quotient of hti by a congruence, and the monoid G ∪{0} for a finite group G. 1. introduction The aim of this paper is to define and study the Hall algebra of the category of set- theoretic representations of a semigroup. Classically, Hall algebras have been studied in the context abelian categories linear over finite fields Fq. Given such a category A, finitary in the sense that Hom(M, N)andExt1(M, N) are finite-dimensional ∀ M, N ∈ A (and therefore finite sets), we may construct from A an associative algebra HA defined 1 over Z , called the Ringel-Hall algebra of A. Asa Z–module, HA is freely generated by the isomorphism classes of objects in A, and its structure constants are expressed in terms of the number of extensions between objects. -
Structure of Some Idempotent Semirings
Journal of Computer and Mathematical Sciences, Vol.7(5), 294-301, May 2016 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online) Structure of Some Idempotent Semirings N. Sulochana1 and T. Vasanthi2 1Assistant Professor, Department of Mathematics, K.S.R.M College of Engineering, Kadapa-516 003, A. P., INDIA. Email: [email protected] 2Professor of Mathematics, Department of Mathematics, Yogi Vemana University, Kadapa-516 003, A. P., INDIA. (Received on: May 30, 2016) ABSTRACT In this paper we study the notion of some special elements such as additive idempotent, idempotent, multiplicatively sub-idempotent, regular, absorbing and characterize these subidempotent and almost idempotent semiring. It is prove that, if S is a subidempotent semiring and (S, ) be a band, then S is a mono semiring and S is k - regular. 2000 Mathematics Subject Classification: 20M10, 16Y60. Keywords: Integral Multiple Property (IMP); Band; Mono Semirings; Rectangular Band; Multiplicatively Subidempotent; k-regular. 1. INTRODUCTION The concept of semirings was first introduced by Vandiver in 1934. However the developments of the theory in semirings have been taking place since 1950. The theory of rings and the theory of semigroups have considerable impact on the developments of the theory of semirings and ordered semirings. It is well known that if the multiplicative structure of an ordered semiring is a rectangular band then its additive structure is a band. P.H. Karvellas2 has proved that if the multiplicative structure of a semiring is a regular semigroup and if the additive structure is an inverse semigroup, then the additive structure is commutative.