scholarly journals On the little secondary Bruhat order

2021 ◽  
Vol 37 (37) ◽  
pp. 113-126
Author(s):  
Rosário Fernandes ◽  
Henrique F. Da Cruz ◽  
Domingos Salomão
Keyword(s):  
Partial Order ◽  
Linear Algebra ◽  
Order Relation ◽  
Bruhat Order ◽  
Symmetric Matrices ◽  
Partial Order Relation ◽  
Minimal Elements ◽  
Cover Relation ◽  
Positive Integers ◽  
Sum Vector

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

scholarly journals Chains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order

10.37236/1143 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Catalin Zara
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Bruhat Order ◽  
Partial Order Relation ◽  
Equivalent Conditions

Let $S_n$ be the group of permutations of $[n]=\{1,\ldots,n\}$. The Bruhat order on $S_n$ is a partial order relation, for which there are several equivalent definitions. Three well-known conditions are based on ascending chains, subwords, and comparison of matrices, respectively. We express the last using fillings of tableaux, and prove that the three equivalent conditions are satisfied in the same number of ways.

scholarly journals Homomorphisms and congruences on ωα-bisimple semigroups

1973 ◽  
Vol 15 (4) ◽  
pp. 441-460 ◽  
Author(s):  
J. W. Hogan
Keyword(s):  
Partial Order ◽  
Ordered Set ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

scholarly journals A study on orderings based on fuzzy quasi- order relations

2021 ◽  
Author(s):  
Zhonglin Chai
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Fuzzy Relation ◽  
Partial Order Relation ◽  
Fuzzy Relations ◽  
Fuzzy Graph ◽  
Feasible Method ◽  
Order Relations ◽  
Finite Set ◽  
Quasi Order

Abstract This paper further studies orderings based on fuzzy quasi-order relations using fuzzy graph. Firstly, a fuzzy relation on a finite set is represented equivalently by a fuzzy graph. Using the graph, some new results on fuzzy relations are derived. In ranking those alternatives, we usually obtain a quasi-order relation, which often has inconsistencies, so it cannot be used for orderings directly. We need to remake it into a reasonable partial order relation for orderings. This paper studies these inconsistencies, and divides them into two types: framework inconsistencies and degree inconsistencies. For the former, a reasonable and feasible method is presented to eliminate them. To eliminate the latter, the concept of complete partial order relation is presented, which is more suitable than partial order relation to rank the alternatives. A method to obtain a reasonable complete partial order relation for a quasi-order relation is given also. An example is given as well to illustrate these discussions. Lastly, the paper discusses the connection between quasi-order relations and preference relations for orderings and some other related problems.

scholarly journals Further Results of the TTT Transform Ordering of Order n

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1960
Author(s):  
Lei Yan ◽  
Diantong Kang ◽  
Haiyan Wang
Keyword(s):  
Reliability Analysis ◽  
Partial Order ◽  
Stochastic Models ◽  
Stochastic Order ◽  
Order Relation ◽  
Partial Order Relation ◽  
Coherent Systems ◽  
Closure Properties ◽  
Distributed Components ◽  
Distribution Class

To compare the variability of two random variables, we can use a partial order relation defined on a distribution class, which contains the anti-symmetry. Recently, Nair et al. studied the properties of total time on test (TTT) transforms of order n and examined their applications in reliability analysis. Based on the TTT transform functions of order n, they proposed a new stochastic order, the TTT transform ordering of order n (TTT-n), and discussed the implications of order TTT-n. The aim of the present study is to consider the closure and reversed closure of the TTT-n ordering. We examine some characterizations of the TTT-n ordering, and obtain the closure and reversed closure properties of this new stochastic order under several reliability operations. Preservation results of this order in several stochastic models are investigated. The closure and reversed closure properties of the TTT-n ordering for coherent systems with dependent and identically distributed components are also obtained.

scholarly journals Complex Stochastic Boolean Systems: Comparing Bitstrings with the Same Hamming Weight

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Luis González
Keyword(s):  
Complex System ◽  
Mathematical Model ◽  
Partial Order ◽  
Order Relation ◽  
Hamming Weight ◽  
Partial Order Relation ◽  
Science And Engineering ◽  
Mutually Independent ◽  
Boolean System

A complex stochastic Boolean system (CSBS) is a complex system depending on an arbitrarily large number n of random Boolean variables. CSBSs arise in many different areas of science and engineering. A proper mathematical model for the analysis of such systems is based on the intrinsic order: a partial order relation defined on the set 0,1n of all binary n-tuples of 0s and 1s. The intrinsic order enables one to compare the occurrence probabilities of two given binary n-tuples with no need to compute them, simply looking at the relative positions of their 0s and 1s. Regarding the analysis of CSBSs, the intrinsic order reduces the complexity of the problem from exponential (2n binary n-tuples) to linear (n Boolean variables). In this paper, using the intrinsic ordering, we compare the occurrence probabilities of any two binary n-tuples having the same number of 1-bits (i.e., the same Hamming weight). Our results can be applied to any CSBS with mutually independent Boolean variables.

Partial order relation for approximation operators in covering based rough sets

2014 ◽  
Vol 284 ◽  
pp. 44-59 ◽  
Author(s):  
Mauricio Restrepo ◽  
Chris Cornelis ◽  
Jonatan Gómez
Keyword(s):  
Partial Order ◽  
Rough Sets ◽  
Order Relation ◽  
Partial Order Relation ◽  
Approximation Operators

scholarly journals Naturally ordered bands

1967 ◽  
Vol 8 (1) ◽  
pp. 55-58 ◽  
Author(s):  
J. M. Howie
Keyword(s):  
Partial Order ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation ◽  
Image Position

In the terminology of Clifford and Preston [2], a band B is a semigroup in which every element is idempotent. On such a semigroup there is a natural (partial) order relation defined by the ruleIf the order relation ≧ is compatible with the multiplication in B, in the sense that e ≧ f and g ≧ h together imply that eg ≧ fh, we shall say that B is a naturally ordered band. The object of this note is to describe the structure of naturally ordered bands.

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