Complete Idempotent Research Articles

THE GREATEST SOLUTION IN THE INEQUALITY OF K X X LX WITH K 2 ISnn;L 2 ISnn;X 2 ISnm ARE A COMPLETE IDEMPOTENT SEMIRINGS OF INTERVAL

The greatest solution of an inequality KX X LX to solve the optimalcontrol problem for P-Temporal Event Graphs, which is to nd the optimal control thatmeets the constraints on the output and constraints imposed on the adjusted model prob-lem (the model matching problem). We give the greatest solution K X X L Xand X H with K; L;X;H matrices whose are entries in a complete idempotent semir-ings. Furthermore, the authors examine the existence of a sucient condition of theprojector in the set of solutions of inequality K X X L X with K; L;X matrixwhose entries are in the complete idempotent semiring. Projectors can be very necessaryto synthesize controllers in manufacturing systems that are constrained by constraintsand some industrial applications. The researcher then examines the requirements forthe presence of the greatest solution was called projector in the set of solutions of theinequality K X X L X with K; L;X matrices whose are entries in an completeidempotent semiring of interval. Researchers describe in more detail the proof of theproperties used to resolve the inequality K X X L X. Before that, we givethe greatest solution of the inequality KX X LX and X G with K; L;X;Gmatrices whose are entries in an complete idempotent semiring of interval

The Singular Value Decomposition over Completed Idempotent Semifields

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.

THE BEST SOLUTION FOR INEQUALITIES OF A O CROSS X LOWER THAN X FROM B O DOT X USING HIGH MATRIX RESIDUATION OF IDEMPOTENT SEMIRING

Abstract A complete idempotent semiring has a structure which is called a complete lattice. Because of the same structure as the complete lattice then inequality of the complete idempotent semiring can be solved a solution by using residuation theory. One of the inequality which is explained is where matrices A,X,B with entries in the complete idempotent semiring S. Furthermore, introduced dual product , i.e. binary operation endowed in a complete idempotent semirings S and not included in the standard definition of complete idempotent semirings. A solution of inequality can be solved by using residuation theory. Because of the guarantee that for each isotone mapping in complete lattice always has a fixed point, then is also exist in a complete idempotent semirings. This of the characteristics is used in order to obtain the greatest solution of inequality . Keywords: complete lattice, complete idempotent semiring, dual Kleene Star, dual product, residuation theory

The spectra of irreducible matrices over completed idempotent semifields

Motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally-ordered semirings, or dioids. We find non-null eigenvectors for every non-null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates. Considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields.

Lemme de Haar sur diverses structures idempotentes. Applications à la comparaison et au contrôle de systèmes à événements discrets

First, we prove two idempotent forms of Haar's lemma corresponding to two main idempotent structures: complete idempotent semirings and semifields. The proof of this lemma is based on residuation theory. Second we show that this lemma plays a central role for two a priori different kinds of problems: the control of discrete event systems (DES) via positive invariance and the comparisons of DES.

Some Relations Between Linear Mappings and Conjugations in Idempotent Analysis

Abstract The theory of lower semicontinuous linear mappings for dioid-valued functions and the abstract (axiomatic) theory of Fenchel-Moreau conjugations have developed parallelly and independently. In this paper we show that these theories are closely related. We introduce the concept of conjugations for functions with values in the top enlargement of a boundedly complete idempotent semifield. Main result: There exists an isomorphism between the complete semimodules of all lower semicontinuous linear mappings and all conjugations. We show that the above framework and the framework of functions with values in the canonical enlargement of a complete lattice ordered group, used in our previous papers to introduce and study abstract conjugations, are essentially equivalent.

A solution to the problem of ( A, B)-invariance for series

The aim of this article is to study a standard problem of control theory, the ( A, B)-invariance problem, which amounts to computing a maximal element X subject to conditions of the form AX⩽ X+ B and X⩽ K. We give a solution to the problem in the framework of formal series over particular complete idempotent semirings. Over finite idempotent semirings, we show that, under the assumption that B and K are recognizable series, the maximal solution exists and is also recognizable. We obtain a similar result for the infinite tropical semiring, with additional hypothesis that the series A is a language, but the notion of recognizable series has to be extended to the weaker notion of pseudo-recognizable series.

Some Relations Between Linear Mappings and Conjugations in Idempotent Analysis

The theory of lower semicontinuous linear mappings for dioid-valued functions and the abstract (axiomatic) theory of Fenchel-Moreau conjugations have developed parallelly and independently. In this paper we show that these theories are closely related. We introduce the concept of conjugations for functions with values in the top enlargement of a boundedly complete idempotent semifield. Main result: There exists an isomorphism between the complete semimodules of all lower semicontinuous linear mappings and all conjugations. We show that the above framework and the framework of functions with values in the canonical enlargement of a complete lattice ordered group, used in our previous papers to introduce and study abstract conjugations, are essentially equivalent.

Pseudovarieties of completely regular semigroups

We shall show that several results concerning the lattice of completely regular semigroup varieties find their analogues for the lattice of pseudovarieties of completely regular semigroups. We establish several complete idempotent endomorphisms and a subdirect decomposition of this lattice of pseudovarieties. These investigations culminate in Theorem 18 which is the analogue for pseudovarieties of Polak’s description [19] of the lattice of completely regular semigroup varieties. We shall in particular be able to describe the lattice of pseudovarieties of orthogroups in terms of the lattice of pseudovarieties of groups.