The intuitionistic fuzzification of the notion of a bi-ideal in ordered semigroups is considered. In terms of intuitionistic fuzzy set, conditions for an ordered semigroup to be completely regular is provided. Characterizations of intuitionistic fuzzy bi-ideals in ordered semigroups are given. Using a collection of bi-ideals with additional conditions, an intuitionistic fuzzy bi-ideal is constr...

We present a new finite set theory implementation for ACL2 wherein sets are implemented as fully ordered lists. This order unifies the notions of set equality and element equality by creating a unique representation for each set, which in turn enables nested sets to be trivially supported and eliminates the need for congruence rules. We demonstrate that ordered sets can be reasoned about in the...

We prove a version of the Doob Meyer decomposition for supermartingales with a linearly ordered index set. Key words Doob Meyer decomposition, natural increasing processes, potentials, supermartingales. Mathematics Subject Classi…cation (2000): : 28A12, 60G07, 60G20.

We show that the problems of deciding whether an ordered set has at least k depth-first linear extensions and whether an ordered set has at least k greedy linear extensions are NP-hard. AMS subject chssi6catiom (1980). 06A10, 68C25.

1 The problem of sorting an intransitive total ordered set, a generalization of regular sorting, is considered. This generalized sorting is based on the fact that there exists a special linear ordering for any intransitive total ordered set. A new data structure called semi-heap is proposed to construct an optimal (n logn) sorting algorithm. Finally, we propose a cost-optimal parallel algorithm...

Reuse of graywater is often suggested to be amogts the most effective to some of the current ecological problems. Application of rough sets theory for discovering dependances between various types of graywater is in the main focus of this work. This can facilitate processes aiming at less energy and chemical use and lower fresh water use. Key–Words: Rough sets, order ideals, quasi-ordered set

Multiple granulation rough set approach to ordered information systems Weihua Xu a b c , Wenxin Sun c , Xiaoyan Zhang c & Wenxiu Zhang d a School of Management, Xi'an Jiaotong University, Xi'an, 710049, P.R. China b The Key Lab of the Ministry of Education for Process Control and Efficiency Engineering, Xi'an, 710049, P.R. China c School of Mathematics and Statistics, Chongqing University of Te...

For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Ze...

In this paper we study a measure, named weakness of an example, which allows us to establish the importance of an example to find representative patterns for the data set editing problem. Our approach consists in reducing the database size without losing information, using algorithm patterns by ordered projections. The idea is to relax the reduction factor with a new parameter, λ, removing all ...

Let P be a finite ordered set, and let J(P ) be the distributive lattice of order ideals of P . The covering relations of J(P ) are naturally associated with elements of P ; in this way, each element of P defines an involution on the set J(P ). Let Γ(P ) be the permutation group generated by these involutions. We show that if P is connected then Γ(P ) is either the alternating or the symmetric ...