# Get e-book Algebraic Structures Using Natural Class of Intervals

Authors in this book introduce a new class of intervals called the natural class of intervals, also known as the special class of intervals or as.

This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. Hence, that an I is not maximal and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A. The dual of an ideal is a filter. The dual of a maximal or prime ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from A to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and can not be proved in ZF , if ZF is consistent.

Within ZF, it is strictly weaker than the axiom of choice. The Ultrafilter Theorem has many equivalent formulations: It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.

Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all clopen sets in some compact totally disconnected Hausdorff topological space. In , the American mathematician Edward V. Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:. Calling 1 , 2 , and 4 a Robbins algebra , the question then becomes: Is every Robbins algebra a Boolean algebra?

This question which came to be known as the Robbins conjecture remained open for decades, and became a favorite question of Alfred Tarski and his students. Every Robbins algebra is a Boolean algebra. For a simplification of McCune's proof, see Dahn Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras".

Generalized Boolean lattices are exactly the ideals of Boolean lattices. A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. However, we have the following property:. Proposition 3 Monotony property: Let us note that the second property is equivalent to the first.

It is its translation in. We can refine the product to become closer to Minkowski's one. It is necessary for some problems to extend the definition of a function defined for real numbers f: It will be convenient to have the same formal expression for f and [f]. Usually the lack of distributivity in Minkowski arithmetic does not give the possibility to get the same formal expressions. But with the pseudo-intervals arithmetic we have presented, there is no data dependency anymore and one can define easily inclusion functions from the natural one. Data dependancy occurs when the variable appears more than once in the function expression.

The deep reason of that is the lack of distributivity in Minkowski arithmetic.

On the other hand, the construction of the inclusion function depends on the type of problem one deals with. This maintains the associativity and distributivity of arithmetic and permits to introduce a pseudo-substraction. The last term corresponds to the transfer of x x - 1. Taylor polynomial expansions, differential calculus and linear algebra operations are defined only in a vector space. Therefore the transfer for the vector space is done directly. This permits to get infinitesimal intervals with the substraction and to compute derivatives.

This is of course not allowed and not possible into the semi-group. This means that [ a , b ] substraction is the anti-interval [- a , - b ] addition. One of the most important consequence is that it is possible to transfer some functions directly to the pseudo-intervals. In order to show that arithmetics and interval algebra developed above are robust and stable, let's try to compute the highest eigenvalue of an interval matrix. One uses here the iterate power method, which is very simple and constitute the basis of several powerful methods such as deflation and others.

The other eigen modes can be computed with the deflation methods for example which consists to withdraw the direction spanned by the eigenvector associated to the highest eigenvalue to the matrix by constructing its projector and to do the same. Several methods are available and efficient to achieve that 6 , We have choosen to compute only the highest eigenvalue and its corresponding eigenvector in order to show simply the efficiency of our new arithmetic. Largest eigenvalue convergence computed with iterate power method to the value computed with scilab Eigenvector components associated to the largest eigenvalue convergence computed with iterate power method to the eigenvector computed with scilab 0.

We would like to use the well-known Schutz-Hotelling algorithm 6 , 35 to inverse the matrix X since it uses simple arithmetic operations:.

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One shows in this section that can be endowed with the metric topology of a Banach space. This permits to define correctly continuity and differentiability of functions. We have to study the two different cases:. So we have a norm on. The normed vector space is a Banach space.

## Vector space

Thus, it is complete. We can consider another equivalent norms on. As is a Banach space, we can describe a notion of differential function on it. This function is continuous and differentiable at any point.

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From the previous example, all monomials are continuous and differentiable, it implies that f is continuous and differentiable as well. This function is not differentiable. This example is very simple but it shows that the result is garanted to be found within the final interval. One sets initial guess interval to [-5, 2], fixed-step of descent to10 -2 , finite difference step to 10 -3 and accuracy of gradient to 10 The results shown in Figures 6 , 7 show that for any initial guess the interval width decreases to converge to real point minimum.

One sets initial guess interval to [0, 10], finite difference step to 10 -3 and accuracy of gradient to10 One can state on Figures 8 , 9 that it finds the same minimum which is an interval centered around We have presented a better algebraic way to do calculations on intervals, called pseudo-intervals vector space or free algebra. It permits to obtain all the basic arithmetic operators with distributivity and associativity.

We have shown that when one increases the representative algebra dimension, the multiplication result will be closer to the usual Minkowski product. It is now possible to build inclusion functions from the natural ones in a systematicway. Thus, it allows to build all algebraic operations and functions on intervals and avoids completely the wrapping effects and data dependance.

One has exhibited some simple examples of applications: This pseudo-intervals arithmetic seems to be a promising and powerful computation framework in several domains of sciences and engineering, such as physics, mechanics of structure, or finance for instance. It is well known that a BL-algebra is an MV-algebra if and only if satisfies the. Also, according to [ 21 ], a residuated lattice is an MV-algebra if and only if satisfies the additional condition.

Let be an MV-algebra. One can see that is an MV-algebra, too. Also the structure of type is called a BCK-algebra if the following axioms are satisfied for all. Then, is a bounded commutative BCK-algebra in which and. Then, is an MV-algebra in which. This set, ordered by inclusion, is a lattice. The meet of two ideals is their intersection and their join is the ideal generated by the union. We define multiplication of two ideals in the usual way: Then, forms a residuated lattice with unit of the ring itself and divisions given by.

It was in this setting that residuated lattices were first defined by Ward and Dilworth [ 23 ]. Then, the unit interval of endowed with the following operations for all becomes an MV-algebra which is called the standard MV-algebra. Also, for each , if we set , , then and are MV-algebras where and.

Let be a residuated lattice and be a nonempty subset of. Trivial examples of filters are and. We will denote by the set of filters. Leustean in [ 24 ] introduced the notion of coannihilator of BL-algebras. Let be a filter of and. The coannihilator of relative to is the set. For any , we will denote by the coannihilator in which is the generated principle ideal of.

Proposition 3 see [ 24 ]. Let and be filters of BL-algebra and. For any nonempty subset of , the coannihilator of is the set. It is easy to see that and. For any subset of , is denoted by. Proposition 4 see [ 24 ]. Let and be two nonempty subsets of BL-algebra , and let be a nonempty family subset of and. In this subsection, we recall some basic notions relevant to soft set.

Let be an initial universe set and let simply denoted by be the set of parameters with respect to. Usually, parameters are attributes, characteristics, or properties of the objects in. The family of all subsets of is denoted by. Definition 5 see [ 6 ].

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A pair is called a soft set over , when , and is a set-valued mapping. In [ 25 ], for a soft set , the set is called the support of the soft set. The soft set is called nonnull if , and it is called a relative null soft set with respect to the parameter set , denoted by, , if. The soft set is called relative whole soft set with respect to the parameter set , denoted by , if , for all,. In the following, for a soft set by we mean the parameterized set. For illustration, Molodtsov considered several examples in [ 6 ]. These examples were also discussed in [ 9 , 11 ].

Now, we give an example of a soft set.

Define on the following operations: One can see that is a divisible residuated lattice. Furthermore, , , but , are incomparable; thus is not a chain. Also, so is not an MTL-algebra. Now, let and which is defined by , where is a permutation on. Then, is a soft set over. Definition 7 see [ 6 ]. Let and be two soft sets over a common universe. Consider divisible residuated lattice in Example 6. Now, we define by and by for each , where , , and. By Proposition 3 , we obtain that. In the following, let be a nonempty family of soft sets over a common universe.

Definition 9 see [ 10 ]. Let be a family of soft sets over a common universe. If , we define and. Let , , and. Now, we define with for each , where. Now, we assume that and. Also, we suppose that in which and in which. By Proposition 3 , we can obtain that , , and , where. Definition 11 see [ 25 ].

### An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices

The cartesian product of is defined as the soft set , where and , for all. Now, we define with. Also, we define with and. Now, we assume that , in which and , in which and , and in which and. Therefore, we have , , and. Definition 13 see [ 10 ]. Let and be two soft sets over a common universe such that. If , we define ii The extended difference of and is defined as the soft set , where , and we have. We suppose that in which and in which and. Then, we can obtain that and. Definition 15 see [ 10 ]. The complement of a soft set over is denoted by and is defined by where is a mapping given by , for all.

In the following, the set of all soft sets over , in which and is a map, is denoted by. Let be a universal set and let be the set of parameters with respect to. One can see that , where and , is a distributive bounded complete lattice if is closed under and. Furthermore, the partial relation defined by lattice operations coincides with. Since is a distributive bounded complete lattice, we can define a new operation as follows: Hence, we obtain the following corollary.