807 the Representation of Boolean Algebras
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES . -
Boolean Like Semirings
Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 13, 619 - 635 Boolean Like Semirings K. Venkateswarlu Department of Mathematics, P.O. Box. 1176 Addis Ababa University, Addis Ababa, Ethiopia [email protected] B. V. N. Murthy Department of Mathematics Simhadri Educational Society group of institutions Sabbavaram Mandal, Visakhapatnam, AP, India [email protected] N. Amarnath Department of Mathematics Praveenya Institute of Marine Engineering and Maritime Studies Modavalasa, Vizianagaram, AP, India [email protected] Abstract In this paper we introduce the concept of Boolean like semiring and study some of its properties. Also we prove that the set of all nilpotent elements of a Boolean like semiring forms an ideal. Further we prove that Nil radical is equal to Jacobson radical in a Boolean like semi ring with right unity. Mathematics Subject Classification: 16Y30,16Y60 Keywords: Boolean like semi rings, Boolean near rings, prime ideal , Jacobson radical, Nil radical 1 Introduction Many ring theoretic generalizations of Boolean rings namely , regular rings of von- neumann[13], p- rings of N.H,Macoy and D.Montgomery [6 ] ,Assoicate rings of I.Sussman k [9], p -rings of Mc-coy and A.L.Foster [1 ] , p1, p2 – rings and Boolean semi rings of N.V.Subrahmanyam[7] have come into light. Among these, Boolean like rings of A.L.Foster [1] arise naturally from general ring dulity considerations and preserve many of the formal 620 K. Venkateswarlu et al properties of Boolean ring. A Boolean like ring is a commutative ring with unity and is of characteristic 2 with ab (1+a) (1+b) = 0. -
Arxiv:2011.04704V2 [Cs.LO] 22 Mar 2021 Eain,Bttre Te Opttoal Neetn Oessuc Models Interesting Well
Domain Semirings United Uli Fahrenberg1, Christian Johansen2, Georg Struth3, and Krzysztof Ziemia´nski4 1Ecole´ Polytechnique, Palaiseau, France 2 University of Oslo, Norway 3University of Sheffield, UK 4University of Warsaw, Poland Abstract Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide. Keywords: semirings, quantales, domain operations 1 Introduction Domain semirings and Kleene algebras with domain [DMS06, DS11] yield particularly simple program ver- ification formalisms in the style of dynamic logics, algebras of predicate transformers or boolean algebras with operators (which are all related). There are two kinds of axiomatisation. Both are inspired by properties of the domain operation on binary relations, but target other computationally interesting models such as program traces or paths on digraphs as well. The initial two-sorted axiomatisation [DMS06] models the domain operation as a map d : S → B from an additively idempotent semiring (S, +, ·, 0, 1) into a boolean subalgebra B of S bounded by 0 and 1. This seems natural as domain elements form powerset algebras in the target models mentioned. Yet the domain algebra B cannot be chosen freely: B must be the maximal boolean subalgebra of S bounded by 0 and 1 and equal to the set Sd of fixpoints of d in S. The alternative, one-sorted axiomatisation [DS11] therefore models d as an endofunction on a semiring S that induces a suitable domain algebra on Sd—yet generally only a bounded distributive lattice. -
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. -
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. -
Mathematics 721 Fall 2016 on Dynkin Systems Definition. a Dynkin
Mathematics 721 Fall 2016 On Dynkin systems Definition. A Dynkin-system D on X is a collection of subsets of X which has the following properties. (i) X 2 D. (ii) If A 2 D then its complement Ac := X n A belong to D. 1 (iii) If An is a sequence of mutually disjoint sets in D then [n=1An 2 D. Observe that every σ-algebra is a Dynkin system. A1. In the literature one can also find a definition with alternative axioms (i), (ii)*, (iii)* where again (i) X 2 D, and (ii)* If A; B are in D and A ⊂ B then B n A 2 D. 1 (iii)* If An 2 D, An ⊂ An+1 for all n = 1; 2; 3;::: then also [n=1An 2 D. Prove that the definition with (i), (ii), (iii) is equivalent with the definition with (i), (ii)*, (iii)*. A2. Verify: If E is any collection of subsets of X then the intersection of all Dynkin-systems containing E is a Dynkin system containing E. It is the smallest Dynkin system containing E. We call it the Dynkin-system generated by E, and denote it by D(E). Definition: A collection A of subsets of X is \-stable if for A 2 A and B 2 A we also have A \ B 2 A. Observe that a \-stable system is stable under finite intersections. A3. (i) Show that if D is a \-stable Dynkin system, then the union of two sets in D is again in D. (ii) Prove: A Dynkin-system is a σ-algebra if and only if it is \-stable. -
Proto-Boolean Rings
An. S¸t. Univ. Ovidius Constant¸a Vol. 19(3), 2011, 175{194 Proto-Boolean rings Sergiu Rudeanu Abstract We introduce the family of R-based proto-Boolean rings associated with an arbitrary commutative ring R. They generalize the proto- Boolean algebra devised by Brown [4] as a tool for expressing in modern language Boole's research in The Laws of Thought. In fact the algebraic results from [4] are recaptured within the framework of proto-Boolean rings, along with other theorems. The free Boolean algebras with a fi- nite or countable set of free generators, and the ring of pseudo-Boolean functions, used in operations research for problems of 0{1 optimization, are also particular cases of proto-Boolean rings. The deductive system in Boole's Laws of Thought [3] involves both an alge- braic calculus and a \general method in Logic" making use of this calculus. Of course, Boole's treatises do not conform to contemporary standards of rigour; for instance, the modern concept of Boolean algebra was in fact introduced by Whitehead [12] in 1898; see e.g. [9]. Several authors have addressed the problem of presenting Boole's creation in modern terms. Thus Beth [1], Section 25 summarizes the approach of Hoff- Hansen and Skolem, who describe the algebra devised by Boole as the quotient of an algebra of polynomial functions by the ideal generated by x2 − x; y2 − y; z2 − z; : : : . Undoubtedly the most comprehensive analysis of the Laws of Thought is that offered by Hailperin [5] in terms of multisets. Quite recently, Brown [4], taking for granted the existence of formal entities described as polynomials with integer coefficients, subject to the usual computation rules 2 except that the indeterminates are idempotent (xi = xi), analyses Chapters V-X of [3] within this elementary framework, which he calls proto-Boolean algebra. -
EXAMPLES of CHAIN DOMAINS 1. Introduction In
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 3, March 1998, Pages 661{667 S 0002-9939(98)04127-6 EXAMPLES OF CHAIN DOMAINS R. MAZUREK AND E. ROSZKOWSKA (Communicated by Ken Goodearl) Abstract. Let γ be a nonzero ordinal such that α + γ = γ for every ordinal α<γ. A chain domain R (i.e. a domain with linearly ordered lattices of left ideals and right ideals) is constructed such that R is isomorphic with all its nonzero factor-rings and γ is the ordinal type of the set of proper ideals of R. The construction provides answers to some open questions. 1. Introduction In [6] Leavitt and van Leeuwen studied the class of rings which are isomorphic with all nonzero factor-rings. They proved that forK every ring R the set of ideals of R is well-ordered and represented by a prime component∈K ordinal γ (i.e. γ>0andα+γ=γfor every ordinal α<γ). In this paper we show that the class contains remarkable examples of rings. Namely, for every prime component ordinalK γ we construct a chain domain R (i.e. a domain whose lattice of left ideals as well as the lattice of right ideals are linearly ordered) such that R and γ is the ordinal type of the set of proper ideals of R. We begin the construction∈K by selecting a semigroup that has properties analogous to those we want R to possess ( 2). The desired ring R is the Jacobson radical of a localization of a semigroup ring§ of the selected semigroup with respect to an Ore set ( 3). -
Free Medial Quandles
Algebra Univers. 78 (2017) 43–54 DOI 10.1007/s00012-017-0443-2 Published online May 23, 2017 © 2017 The Author(s) Algebra Universalis This article is an open access publication Free medial quandles Premyslˇ Jedlicka,ˇ Agata Pilitowska, and Anna Zamojska-Dzienio Abstract. This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles. 1. Introduction A binary algebra (Q, ) is called a rack if the following conditions hold, for · every x, y, z Q: ∈ x(yz)=(xy)(xz) (we say Q is left distributive), • the equation xu = y has a unique solution u Q (we say Q is a left • ∈ quasigroup). An idempotent rack is called a quandle (we say Q is idempotent if xx = x for every x Q). A quandle Q is medial if, for every x, y, u, v Q, ∈ ∈ (xy)(uv)=(xu)(yv). An important example of a medial quandle is an abelian group A with an operation defined by a b = (1 h)(a)+h(b), where h is an automorphism ∗ ∗ − of A. This construction is called an affine quandle (or sometimes an Alexander quandle) and denoted by Aff(A, h). In the literature [6, 7], the group A is 1 usually considered to be a Z[t, t− ]-module, where t a = h(a), for each a A. · ∈ We shall adopt this point of view here as well and we usually write Aff(A, r) instead, where r is a ring element. Note that in universal algebra terminology, an algebra is said to be affine if it is polynomially equivalent to a module. -
Unique Factorization Monoids and Domains
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 2, May 1971 UNIQUE FACTORIZATION MONOIDS AND DOMAINS R. E. JOHNSON Abstract. It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e., whenever b <a and aMp\bM¿¿0, then aMQbM). By a monoid, we shall always mean a cancellative semigroup with identity element. All rings are assumed to have unities, and integral domains and fields are not assumed to be commutative. If R is an integral domain, then the multiplicative monoid of nonzero elements is denoted by i?x. For any ring or monoid A, U(A) will denote its group of units. A monoid M is called a unique factorization monoid (uf-monoid) iff for every mEM — U(M), any two factorizations of m have a common refinement. That is, if m=aia2 ■ ■ ■ ar = bib2 ■ ■ ■ bs, then the a's and 6's can be factored, say ai = ci • • • c,, a2 = a+i ■ ■ ■Cj, ■ ■ ■ , 61 = di ■ ■ ■dk, b2=dk+i ■ ■ • dp, ■ ■ ■ , in such a way that cn=dn for all n. An integral domain R is called a unique factorization domain (uf-domain) iff Rx is a uf-monoid. A monoid M with relation < is said to be ordered by < iff < is a transitive linear ordering such that whenever a<b then ac<bc and ca<cb for all cEM. -
On Strategy-Proofness and Semilattice Single-Peakedness Jordi Massó April 2019
On Strategy-Proofness and Semilattice Single-Peakedness Jordi Massó April 2019 Barcelona GSE Working Paper Series Working Paper nº 1087 On strategy-proofness and semilattice single-peakedness∗ Agust´ın G. Bonifacio† Jordi Mass´o‡ April 11, 2019 Abstract We study social choice rules defined on the domain of semilattice single- peaked preferences. Semilattice single-peakedness has been identified as the necessary condition that a set of preferences must satisfy so that the set can be the domain of a strategy-proof, tops-only, anonymous and unanimous rule. We characterize the class of all such rules on that domain and show that they are deeply related to the supremum of the underlying semilattice structure. Keywords: Strategy-proofness; Unanimity; Anonymity; Tops-onlyness; Single-peakedness. JEL Classification Numbers: D71. 1 Introduction We characterize the class of all strategy-proof and simple rules defined on the domain of semilattice single-peaked preferences. A rule is a systematic procedure to select ∗We thank Salvador Barber`a, Shurojit Chatterji and Bernardo Moreno for helpful comments and suggestions. Bonifacio acknowledges financial support received from the UNSL, through grant 319502, and from the Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), through grant PIP 112-200801-00655. Mass´oacknowledges financial support received from the Span- ish Ministry of Economy and Competitiveness, through grant ECO2017-83534-P and through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563), and from the Gener- alitat de Catalunya, through grant SGR2017-711. †Instituto de Matem´atica Aplicada San Luis, Universidad Nacional de San Luis and CONICET, San Luis, Argentina; e-mail: [email protected] ‡Departament d’Economia i d’Hist`oria Econ`omica, Universitat Aut`onoma de Barcelona, and Barcelona GSE, Spain; e-mail: [email protected] 1 an alternative as a function of the declared profile of agents’ preferences.