Relations and functions in multiset context

Girish, K. P.; John, Sunil Jacob

doi:10.1016/j.ins.2008.11.002

Cited by 66 publications

(41 citation statements)

References 12 publications

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“…Beeri et al [4] show that any lattice L can be encoded using a minimal Armstrong relation which contains at least dð1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8:jMðLÞj p Þ=2e rows and at most jMðLÞj þ 1 rows. Multiset relations which are defined as a sub multiset of the generalized Cartesian product [14] can be considered in the future.…”

Section: Resultsmentioning

confidence: 99%

Representing lattices using many-valued relations

Gély¹,

Medina²,

Nourine³

2009

Information Sciences

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Section: Resultsmentioning

confidence: 99%

Representing lattices using many-valued relations

Gély¹,

Medina²,

Nourine³

2009

Information Sciences

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“…In this section, a brief survey of results and notations as introduced in literature() is presented.…”

Section: Preliminaries and Basic Definitionsmentioning

confidence: 99%

Multiset topology via DNA and RNA mutation

El-Sharkasy

Fouda

Badr

2018

Math Methods in App Sciences

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Topology is the most important branch of modern mathematics, which plays an important role in applications. In this paper, we use the concept of the topology, based on the concept of multiset to solve an important problems in life (DNA and RNA mutation) to detect diseases and help biologists in the treatment of diseases. Also, we introduce a new theory that explains if there is an existence of a mutation or not, and we have a set of metric functions through which we examine the congruence and similarity and dissimilarity between "types," which may be a strings of bits, vectors, DNA or RNA sequences, … , etc. Finally, we will introduce a theory that which can be used to know the existence and place of mutation.In this section, a brief survey of results and notations as introduced in literature 7-15 is presented. 5820 Definition 2.1. A mset M drawn from the set X is represented by a mapping C M (x) defined as C M (x) ∶ X → N, where N is the set of positive integers. C M (x) is the number of occurrences of the element x in the mset M. We present the mset M drawn from the set X = {x 1 , x 2 , … , x n } as M = {m 1 ∕x 1 , m 2 ∕x 2 , … , m n ∕x n } where m i is the number of occurrences of the element x i , i = 1, 2, … , n, in the mset M . Clearly, a set is a special case of an mset.Let M and N be 2 msets drawn from a set X. Then, the following are defined. 6,13,[16][17][18] The cardinality of an mset M is symbolized by Card (M) or |M| and is given byDefinition 2.2. (Girish and John 12 and Jena et al 19 ; whole submsets) A submset N of M is a whole submset of M with each element in N having full multiplicity as in M, ie, C N (x) = C M (x) for every x in N. Definition 2.3. (Girish and John 12 and Jena et al 19 ; partial whole submsets) A submset N of M is a partial whole submset of M with at least one element in N having full multiplicity as in M, ie, C N (x) = C M (x) for some x in N. Definition 2.4. (Girish and John 12 and Jena et al 19 ; full submsets) A submset N of M is a full submset of M if M and N having the same support set with C N (x) ≤ C M (x) for every x in N, ie, M * = N * with C N (x) ≤ C M (x) for every x in N. Example 2.5. (Girish and John 12 ) Let M = {3∕a, 4∕b, 6∕c} be an mset. The following are some of submsets of M that are whole submsets, partial whole submsets, and full submsets: (a) A submset {3∕a, 4∕b} is a whole submset and partial whole submset of M. (b) A submset {2∕a, 4∕b, 3∕c} is a partial whole submset and full submset of M. (c) A submset {2∕a, 4∕b} is a partial whole submset of M. Definition 2.6. (Previous studies 9-12 ) Let M 1 and M 2 be 2 msets drawn from a set Y; then the Cartesian product of M 1 and M 2 is defined as M 1 × M 2 = {(m∕x, n∕ )∕mn ∶ x∈ m M 1 , ∈ n M 2 }. Definition 2.7. (Girish and John 9 and Girish and Sunil 10 ) A submset R of M × M is called an mset relation on M if every member (m∕x, n∕y) of R has a count and product of C 1 (x, y) and C 2 (x, y), We denote m∕x related to n∕y by m∕xRn∕y. Definition 2.8. (Girish and John 9 and Girish and Sunil 10 ) (i) An mset relation...

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“…Then R.R.Yager [6] introduced the concept of fuzzy multi set which are useful for handling Problems With multi dimensional characterization properties and T.V.Ramakrishnan and S.Sabu [9] proposed fuzzy multi sets in 2010.In 1991, A.S.Binshahan [11] introduced and investigated the notations of fuzzy pre-open and fuzzy pre-closed sets in 2003, T.Fukutake, R.K.Saraf, M.Caldas and S.Mishra introduced fuzzy generalized pre-closed sets in fuzzy topological space.T.Shinoj and sunil Jacob john [10] proposed Intuitionistic fuzzy multi set (IFMS) in 2012 which is the combination of intuitionstic fuzzy set and fuzzy multi set. P.Rajarajeswari and Krishnamoorthy [11] introduced the concept of Intuitionistic fuzzy weakly generalized closed set.…”

Section: Introductionmentioning

confidence: 98%

On Intuitionistic Fuzzy Multi Generalized Pre-Closed Set

Sakthivel¹,

Prabha²,

Gayathriprabha³

2016

IJCA

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The purpose of this paper is to introduce and study the concept of Intuitionistic fuzzy multi generalized pre-closed set and Intuitionistic fuzzy multi generalized pre-open set in Intuitionistic fuzzy multi topological space and investigate some of its properties. KeywordsIntuitionistic fuzzy multi topology, Intuitionistic fuzzy multi generalized pre closed set , Intuitionistic fuzzy multi generalized pre open set.

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Relations and functions in multiset context

Cited by 66 publications

References 12 publications

Representing lattices using many-valued relations

Representing lattices using many-valued relations

Multiset topology via DNA and RNA mutation

On Intuitionistic Fuzzy Multi Generalized Pre-Closed Set

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