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The Linear Discrepancy of a Fuzzy Poset
Cheong, Min-Seok,Chae, Gab-Byung,Kim, Sang-Mok Korean Institute of Intelligent Systems 2011 INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGE Vol.11 No.1
In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X ${\times}$ X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the 'less than' function, G is the 'greater than' function, and I is the 'incomparable to' function. Using this approach, we are able to define a special class of fuzzy posets, and define the 'skeleton' of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)${\mid}$f(x)-f(y)${\mid}$ for f ${\in}$ F and x, y ${\in}$ X with I(x, y) > $\frac{1}{2}$, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.
나인혁,정민석,김상목 장전수학회 2015 Proceedings of the Jangjeon mathematical society Vol.18 No.1
In this paper, we deal with centralities in partially ordered sets. We rst dene the center and an eccentricity on a poset. And then, using these, we divide its ground set into three useful subposets PI , PU, and PD of a poset P with respect to the position of the farthest elements from each poset element, respectively. Next, we give some criteria for determining whether an element of P belongs to PI , PU, or PD by using relations between the subposets and the eccentricity. Finally, we obtain some necessary conditions for being a center of a lattice in terms of PI , PU, and PD.
The Linear Discrepancy of a Fuzzy Poset
Minseok Cheong,Gab-Byung Chae,Sang-Mok Kim 한국지능시스템학회 2011 INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGE Vol.11 No.1
In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X × X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the ‘less than’ function, G is the ‘greater than’ function, and I is the ‘incomparable to’ function. Using this approach, we are able to define a special class of fuzzy posets, and define the ‘skeleton’ of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)|f(x)?f(y)| for f ∈ F and x, y ∈ X with I(x, y) > ½, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.
Generating sets of strictly order-preserving transformation semigroups on a finite set
Hayrullah Ayik,Leyla Bugay 대한수학회 2014 대한수학회보 Vol.51 No.4
Let On and POn denote the order-preserving transformation and the partial order-preserving transformation semigroups on the set Xn = {1, . . . , n}, respectively. Then the strictly partial order-preserving transformation semigroup SPOn on the set Xn, under its natural or- der, is defined by SPOn = POn \ On. In this paper we find necessary and sufficient conditions for any subset of SPO(n, r) to be a (minimal) generating set of SPO(n, r) for 2 r n − 1.
GENERATING SETS OF STRICTLY ORDER-PRESERVING TRANSFORMATION SEMIGROUPS ON A FINITE SET
Ayik, Hayrullah,Bugay, Leyla Korean Mathematical Society 2014 대한수학회보 Vol.51 No.4
Let $O_n$ and $PO_n$ denote the order-preserving transformation and the partial order-preserving transformation semigroups on the set $X_n=\{1,{\ldots},n\}$, respectively. Then the strictly partial order-preserving transformation semigroup $SPO_n$ on the set $X_n$, under its natural order, is defined by $SPO_n=PO_n{\setminus}O_n$. In this paper we find necessary and sufficient conditions for any subset of SPO(n, r) to be a (minimal) generating set of SPO(n, r) for $2{\leq}r{\leq}n-1$.
The Linear Discrepancy of a Fuzzy Poset
정민석,채갑병,김상목 한국지능시스템학회 2011 INTERNATIONAL JOURNAL of FUZZY LOGIC and INTELLIGE Vol.11 No.1
In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X × X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the ‘less than’function, G is the ‘greater than’ function, and I is the ‘incomparable to’ function. Using this approach, we are able to define a special class of fuzzy posets, and define the ‘skeleton’ of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)|f(x)−f(y)| for f ∈ F and x, y ∈ X with I(x, y) > ½ , where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.
G. V. R. BABU,P. Subhashini,P. D. Sailaja 장전수학회 2013 Advanced Studies in Contemporary Mathematics Vol.23 No.3
The purpose of this paper is to extend the fixed point results of maps satisfying weak contractivity condition introduced by Bhaskar and Lakshmikantam [T. Gnana Bhaskar and V. Lakshmikantham, Fixed point theorem in partially ordered metric spaces and applications, Non-linear Analysis, 65 (2006), 1379-1393] to the case of contractions with rational expressions in partially ordered metric spaces. Examples are provided in support of our results. Our theorems generalize the results of Bhaskar and Lakshmikantam [2].
CHOUDHURY, BINAYAK S.,KONAR, PULAK,METIYA, NIKHILESH The Korean Society for Computational and Applied M 2019 Journal of applied mathematics & informatics Vol.37 No.1
In this paper we prove certain coupled coincidence point and coupled common fixed point results in partially ordered metric spaces for a pair of compatible mappings which satisfy certain rational inequality. The results are supported with two examples.
Simple posets with respect to linear Discrepancies
정민석,채갑병,김상목 장전수학회 2013 Advanced Studies in Contemporary Mathematics Vol.23 No.1
In this paper, we give a characterization of n-posets with respect to linear discrepancy. We rst obtain the uniquely dened tri-angular matrix r lled with 1 from the interchanging operation on the matrix representation of naturally labeled posets and their linear dis-crepancies. Next, for each linear discrepancy l and each cardinality of poset n, we dene the (n; l)-simple poset S with the property that a removal of any order relation from S causes its linear discrepancy in-creasing. Then, in terms of poset extension of (n; l)-simple posets, we give a characterization of n-posets with respect to linear discrepancy l, by using the fact that (n; l)-simple posets are represented as rs. Lastly, as an application of simple posets represented by rs, we count all n-posets of linear discrepancy 1, which is identical to the previous result due to Tanenbaum, et al. [Linear discrepancy and weak discrepancy of partially ordered sets, Order 18 (2001), 201-225].
BINAYAK S. CHOUDHURY,Pulak Konar,Nikhilesh Metiya 한국전산응용수학회 2019 Journal of applied mathematics & informatics Vol.37 No.1
In this paper we prove certain coupled coincidence point and coupled common xed point results in partially ordered metric spaces for a pair of compatible mappings which satisfy certain rational inequality. The results are supported with two examples.