Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations Under Decisiveness
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Relations on Semigroups
International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Relations on Semigroups 1D.D.Padma Priya, 2G.Shobhalatha, 3U.Nagireddy, 4R.Bhuvana Vijaya 1 Sr.Assistant Professor, Department of Mathematics, New Horizon College Of Engineering, Bangalore, India, Research scholar, Department of Mathematics, JNTUA- Anantapuram [email protected] 2Professor, Department of Mathematics, SKU-Anantapuram, India, [email protected] 3Assistant Professor, Rayalaseema University, Kurnool, India, [email protected] 4Associate Professor, Department of Mathematics, JNTUA- Anantapuram, India, [email protected] Abstract: Equivalence relations play a vital role in the study of quotient structures of different algebraic structures. Semigroups being one of the algebraic structures are sets with associative binary operation defined on them. Semigroup theory is one of such subject to determine and analyze equivalence relations in the sense that it could be easily understood. This paper contains the quotient structures of semigroups by extending equivalence relations as congruences. We define different types of relations on the semigroups and prove they are equivalence, partial order, congruence or weakly separative congruence relations. Keywords: Semigroup, binary relation, Equivalence and congruence relations. I. INTRODUCTION [1,2,3 and 4] Algebraic structures play a prominent role in mathematics with wide range of applications in science and engineering. A semigroup -
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. -
Binary Relations and Functions
Harvard University, Math 101, Spring 2015 Binary relations and Functions 1 Binary Relations Intuitively, a binary relation is a rule to pair elements of a sets A to element of a set B. When two elements a 2 A is in a relation to an element b 2 B we write a R b . Since the order is relevant, we can completely characterize a relation R by the set of ordered pairs (a; b) such that a R b. This motivates the following formal definition: Definition A binary relation between two sets A and B is a subset of the Cartesian product A×B. In other words, a binary relation is an element of P(A × B). A binary relation on A is a subset of P(A2). It is useful to introduce the notions of domain and range of a binary relation R from a set A to a set B: • Dom(R) = fx 2 A : 9 y 2 B xRyg Ran(R) = fy 2 B : 9 x 2 A : xRyg. 2 Properties of a relation on a set Given a binary relation R on a set A, we have the following definitions: • R is transitive iff 8x; y; z 2 A :(xRy and yRz) =) xRz: • R is reflexive iff 8x 2 A : x Rx • R is irreflexive iff 8x 2 A : :(xRx) • R is symmetric iff 8x; y 2 A : xRy =) yRx • R is asymmetric iff 8x; y 2 A : xRy =):(yRx): 1 • R is antisymmetric iff 8x; y 2 A :(xRy and yRx) =) x = y: In a given set A, we can always define one special relation called the identity relation. -
Relations 21
2. CLOSURE OF RELATIONS 21 2. Closure of Relations 2.1. Definition of the Closure of Relations. Definition 2.1.1. Given a relation R on a set A and a property P of relations, the closure of R with respect to property P , denoted ClP (R), is smallest relation on A that contains R and has property P . That is, ClP (R) is the relation obtained by adding the minimum number of ordered pairs to R necessary to obtain property P . Discussion To say that ClP (R) is the “smallest” relation on A containing R and having property P means that • R ⊆ ClP (R), • ClP (R) has property P , and • if S is another relation with property P and R ⊆ S, then ClP (R) ⊆ S. The following theorem gives an equivalent way to define the closure of a relation. \ Theorem 2.1.1. If R is a relation on a set A, then ClP (R) = S, where S∈P P = {S|R ⊆ S and S has property P }. Exercise 2.1.1. Prove Theorem 2.1.1. [Recall that one way to show two sets, A and B, are equal is to show A ⊆ B and B ⊆ A.] 2.2. Reflexive Closure. Definition 2.2.1. Let A be a set and let ∆ = {(x, x)|x ∈ A}. ∆ is called the diagonal relation on A. Theorem 2.2.1. Let R be a relation on A. The reflexive closure of R, denoted r(R), is the relation R ∪ ∆. Proof. Clearly, R ∪ ∆ is reflexive, since (a, a) ∈ ∆ ⊆ R ∪ ∆ for every a ∈ A. -
Scalability of Reflexive-Transitive Closure Tables As Hierarchical Data Solutions
Scalability of Reflexive- Transitive Closure Tables as Hierarchical Data Solutions Analysis via SQL Server 2008 R2 Brandon Berry, Engineer 12/8/2011 Reflexive-Transitive Closure (RTC) tables, when used to persist hierarchical data, present many advantages over alternative persistence strategies such as path enumeration. Such advantages include, but are not limited to, referential integrity and a simple structure for data queries. However, RTC tables grow exponentially and present a concern with scaling both operational performance of the RTC table and volume of the RTC data. Discovering these practical performance boundaries involves understanding the growth patterns of reflexive and transitive binary relations and observing sample data for large hierarchical models. Table of Contents 1 Introduction ......................................................................................................................................... 2 2 Reflexive-Transitive Closure Properties ................................................................................................. 3 2.1 Reflexivity ...................................................................................................................................... 3 2.2 Transitivity ..................................................................................................................................... 3 2.3 Closure .......................................................................................................................................... 4 3 Scalability -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
MATH 213 Chapter 9: Relations
MATH 213 Chapter 9: Relations Dr. Eric Bancroft Fall 2013 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R ⊆ A×B, i.e., R is a set of ordered pairs where the first element from each pair is from A and the second element is from B. If (a; b) 2 R then we write a R b or a ∼ b (read \a relates/is related to b [by R]"). If (a; b) 2= R, then we write a R6 b or a b. We can represent relations graphically or with a chart in addition to a set description. Example 1. A = f0; 1; 2g;B = f1; 2; 3g;R = f(1; 1); (2; 1); (2; 2)g Example 2. (a) \Parent" (b) 8x; y 2 Z; x R y () x2 + y2 = 8 (c) A = f0; 1; 2g;B = f1; 2; 3g; a R b () a + b ≥ 3 1 Note: All functions are relations, but not all relations are functions. Definition 2. If A is a set, then a relation on A is a relation from A to A. Example 3. How many relations are there on a set with. (a) two elements? (b) n elements? (c) 14 elements? Properties of Relations Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if a R a for all a 2 A. Definition 4 (Symmetric). A relation R on a set A is said to be symmetric if and only if a R b =) b R a for all a; b 2 A. -
Semiring Orders in a Semiring -.:: Natural Sciences Publishing
Appl. Math. Inf. Sci. 6, No. 1, 99-102 (2012) 99 Applied Mathematics & Information Sciences An International Journal °c 2012 NSP Natural Sciences Publishing Cor. Semiring Orders in a Semiring Jeong Soon Han1, Hee Sik Kim2 and J. Neggers3 1 Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea 2 Department of Mathematics, Research Institute for Natural Research, Hanyang University, Seoal, Korea 3 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U.S.A Received: Received May 03, 2011; Accepted August 23, 2011 Published online: 1 January 2012 Abstract: Given a semiring it is possible to associate a variety of partial orders with it in quite natural ways, connected with both its additive and its multiplicative structures. These partial orders are related among themselves in an interesting manner is no surprise therefore. Given particular types of semirings, e.g., commutative semirings, these relationships become even more strict. Finally, in terms of the arithmetic of semirings in general or of some special type the fact that certain pairs of elements are comparable in one of these orders may have computable and interesting consequences also. It is the purpose of this paper to consider all these aspects in some detail and to obtain several results as a consequence. Keywords: semiring, semiring order, partial order, commutative. The notion of a semiring was first introduced by H. S. they are equivalent (see [7]). J. Neggers et al. ([5, 6]) dis- Vandiver in 1934, but implicitly semirings had appeared cussed the notion of semiring order in semirings, and ob- earlier in studies on the theory of ideals of rings ([2]). -
On Tarski's Axiomatization of Mereology
On Tarski’s Axiomatization of Mereology Neil Tennant Studia Logica An International Journal for Symbolic Logic ISSN 0039-3215 Volume 107 Number 6 Stud Logica (2019) 107:1089-1102 DOI 10.1007/s11225-018-9819-3 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Nature B.V.. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Neil Tennant On Tarski’s Axiomatization of Mereology Abstract. It is shown how Tarski’s 1929 axiomatization of mereology secures the re- flexivity of the ‘part of’ relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant rea- soning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness. Keywords: Mereology, Part of, Reflexivity, Tarski, Axiomatization, Constructivity. -
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.