In this paper the chain length of a space of fuzzy orderings is defined, and various properties of this invariant are proved. The structure theorem for spaces of finite chain length is proved. Spaces of Fuzzy Orderings Throughout X = (X,A) denoted a space of fuzzy orderings. That is, A is a fuzzy subgroup of abelian group G of exponent 2. (see [1] (i.e. x 2 = 1,  x  G), and X is a (non empty) fuzzy subset of the character group ï£ (A) = Hom(A,{1,–1}) satisfying: 1. X is a fuzzy closed subset of ï£ (A). 2.  an element e  A such that ï³(e) = – 1  ï³ ïƒŽ X. 3. Xïž :={a  A\ ï³(a) = 1  ï³ ïƒŽ X} = 1. 4. If f and g are forms over A and if x  D( f  g) then  y  D( f ) and z  D(g) such that x  D<y, z >. Observe, by 3, that the element e  A whose existence is asserted by 2 is unique. Also, e  1 (since ï³(1) = 1  ï³ ïƒŽ X). Notice that for a  A, the set X(a):= {ï³ ïƒŽ Xï³(a) = 1} is clopen (i.e. both closed and open) in X. Moreover, ï³(a) = – 1  ï³(– a) = 1 holds for any ï³ ïƒŽ X (by 2).
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In this paper, Min-Max composition fuzzy relation equation are studied. This study is a generalization of the works of Ohsato and Sekigushi. The conditions for the existence of solutions are studied, then the resolution of equations is discussed.
In thisˑ paperˑ, we apply the notion ofˑ intuitionisticˑ fuzzyˑ n-fold KU-ideal of KU-algebra. Some types of ideals such as intuitionistic fuzzy KU-ideal, intuitionisticˑ fuzzy closed idealˑ and intuitionistic fuzzy n-fold KU-ideal are studied. Also, the relations between intuitionistic fuzzy n-fold KU-ideal and intuitionistic fuzzy KU-ideal are discussed. Furthermore, aˑ fewˑ results of intuitionisticˑ fuzzyˑ n-ˑfold KU-ideals of a KU-algebra underˑ homomorphismˑ are discussed.
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In this paper, we apply the notion of a bipolar fuzzy n-fold KU-ideal of KU- algebras. We introduce the concept of a bipolar fuzzy n-fold KU-ideal and investigate several properties. Also, we give relations between a bipolar fuzzy n- fold KU-ideal and n-fold KU-ideal. The image and the pre-image of bipolar fuzzy n-fold KU-ideals in KU-algebras are defined and how the image and the pre- image of bipolar fuzzy n-fold KU-ideals in KU-algebras become bipolar fuzzy n- fold KU-ideals are studied. Moreover, the product of bipolar fuzzy n-fold KU- ideals in Cartesian product KU-algebras is given.
The last decade of this 20th century provides a wide spread of applications of one of the computer techniques, which is called Fuzzy Logic. This technique depends mainly on the fuzzy set theory, which is considered as a general domain with respect to the conventional set theory. This paper presents in initiative the fuzzy sets theory and fuzzy logic as a complete mathematics system. Here it was explained the concept of fuzzy set and defined the operations of fuzzy logic. It contains eleven operations beside the other operations which related to fuzzy algebra. Such search is considered as an enhancement for supporting the others waiting search activities in this field.
This paper introduces the concept of fuzzy σ-ring as a generalization of fuzzy σ-algebra and basic properties; examples of this concept have been given. As the first result, it has been proved that every σ-algebra over a fuzzy set x* is a fuzzy σ-ring-over a fuzzy set x* and construct their converse by example. Furthermore, the fuzzy ring concept has been studied to generalize fuzzy algebra and its relation. Investigating that the concept of fuzzy σ-Ring is a stronger form of a fuzzy ring that is every fuzzy σ-Ring over a fuzzy set x* is a fuzzy ring over a fuzzy set x* and construct their converse by example. In addition, the idea of the smallest, as an important property in the study of real analysis, is studied
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Both traditional and novel techniques were employed in this work for magnetic shielding evaluation to shed new light on the magnetic and aromaticity properties of benzene and 12 [n]paracyclophanes with n = 3–14. Density functional theory (DFT) with the B3LYP functional and all-electron Jorge-ATZP and x2c-TZVPPall-s basis sets was utilized for geometry optimization and magnetic shielding calculations, respectively. Additionally, the 6-311+G(d,p) basis set was incorporated for the purpose of comparing the magnetic shielding results. In addition to traditional evaluations such as NICS/NICSzz-Scan, and 2D-3D σiso(r)/σzz(r) maps, two new techniques were implemented: bendable grids (BGs) and cylindrical grids (CGs) of ghost atoms (Bqs). BGs a
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Transportation and distribution are the most important elements in the work system for any company, which are of great importance in the success of the chain work. Al-Rabee factory is one of the largest ice cream factories in Iraq and it is considered one of the most productive and diversified factories with products where its products cover most areas of the capital Baghdad, however, it lacks a distribution system based on scientific and mathematical methods to work in the transportation and distribution processes, moreover, these processes need a set of important data that cannot in any way be separated from the reality of fuzziness industrial environment in Iraq, which led to use the fuzzy sets theory to reduce the levels of uncertainty.
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In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.
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