Domain Semirings United

Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.

Subdirect products of an idempotent semiring and a b-lattice of skew-rings

Bandelt and Petrich [Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. (2) 25(2) (1982) 155–171] characterized a class of additive inverse semirings which are subdirect products of a distributive lattice and a ring. The aim of this paper is to characterize a class of additively regular semirings which are subdirect products of an idempotent semiring and a [Formula: see text]-lattice of skew-rings.

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

Derivations of matrix semirings

It is well known that if [Formula: see text] is a derivation in semiring [Formula: see text], then in the semiring [Formula: see text] of [Formula: see text] matrices over [Formula: see text], the map [Formula: see text] such that [Formula: see text] for any matrix [Formula: see text] is a derivation. These derivations are used in matrix calculus, differential equations, statistics, physics and engineering and are called hereditary derivations. On the other hand (in sense of [Basic Algebra II (W. H. Freeman & Company, 1989)]) [Formula: see text]-derivation in matrix semiring [Formula: see text] is a [Formula: see text]-linear map [Formula: see text] such that [Formula: see text], where [Formula: see text]. We prove that if [Formula: see text] is a commutative additively idempotent semiring any [Formula: see text]-derivation is a hereditary derivation. Moreover, for an arbitrary derivation [Formula: see text] the derivation [Formula: see text] in [Formula: see text] is of a special type, called inner derivation (in additively, idempotent semiring). In the last section of the paper for a noncommutative semiring [Formula: see text] a concept of left (right) Ore elements in [Formula: see text] is introduced. Then we extend the center [Formula: see text] to the semiring LO[Formula: see text] of left Ore elements or to the semiring RO[Formula: see text] of right Ore elements in [Formula: see text]. We construct left (right) derivations in these semirings and generalize the result from the commutative case.

Derivations of polynomial semirings

The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring [Formula: see text], where [Formula: see text] is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal [Formula: see text] and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider [Formula: see text]-derivations in classical sense of Jacobson. In the main result, Theorem 5.3, we show that any derivation in [Formula: see text] can be represented as a sum of a generalized inner derivation and an [Formula: see text]-derivation.

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

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

Maximal k-ideals and r-ideals in semirings

Some properties of (left) k-ideals and r-ideals of a semiring are considered by the help of the congruence class semiring. It is proved that a proper k-ideal of a semiring with an identity is prime if it is a maximal left k-ideal. An equivalent condition for a proper r-ideal of a semiring being a maximal (left) r-ideal is established. It is shown that (left) r-ideals and (left) k-ideals coincide for an additively idempotent semiring, though the former is a special kind of the latter in general. It is proved that a proper k-ideal of an incline with an identity is a maximal k-ideal if and only if the corresponding congruence class semiring is the Boolean semiring.

An Algorithm of Plus-Closures of Loop-Nonnegative Matrices over Idempotent Semirings and its Applications

To judge the loop-nonnegativity of a matrix over an idempotent semiring and compute the plus-closure of when it is loop-nonnegative, a Plus_Closure_of_Matrix algorithm of complexity is constructed and proved. As a generalization of Floyd algorithm, Warshall algorithm as well as Gauβ-Jordan Elimination algorithm on idempotent semirings, this algorithm can also be used to solve some Algebraic Path Problems, Shortest Path Problems and the transitive closures of matrices over idempotent semirings even if the idempotent semirings have no completeness and closeness.