满足a+ab=a+b的幂等半环的结构

本文讨论了满足a+ab=a+b的幂等半环的结构,给出这种幂等半环是左零半环的伪强右正规幂等半环,并得出这种幂等半环与环的直积是左环的伪强右正规幂等半环.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

Idempotent distributive semirings II

In [2] it is shown that every idempotent distributive semiring is the P?onka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.     

Idempotent semirings with a commutative additive reduct

For every semigroup S , we define a congruence relation ρ on the power semiring (P(S),\cup,\circ) of S . If S is a band, then P(S)/ρ is an idempotent semiring . This enables us to find models for the free objects in the variety of idempotent semiring s whose additive reduct is a semilattice. December 28, 1999     

Fuzzy semirings

In this paper we initiate the study of fuzzy semirings and fuzzy A-semimodules where A is a semiring and A-semimodules are representations of A. In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring A is fully idempotent if and only if the lattice of fuzzy ideals of A is distributive under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring A admits the structure of a topological space, called the fuzzy prime spectrum of A.     

On quasi completely regular semirings

We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.     

A remark on nonfinitely based semirings

We generalize a criterion from a previous paper which ensures that an additively idempotent semiring is not finitely based. As a consequence, we prove the NFB property for the semiring generated by transformations on a finite set with more than one element. Supported by Grant No. 144011 of the Ministry of Science of the Republic of Serbia.     

某些半环上Green关系的刻划

假设S是乘法半群为完全正则半群的半环.给出了S上的Green关系H,L和D是S上的半环同余的等价刻划,并利用幂等元的方法证明了在一定条件下D是S上的同余当且仅当L,R是S上的同余.     

Varieties of idempotent semirings with commutative addition

The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13 distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings free in these varieties are given. Received April 22, 2004; accepted in final form June 3, 2005.     

一类幂等半环的性质和结构

研究了加法半群为半格的半环类S+l中的乘法带半环和矩形带半环类BR中的乘法带半环;给出了ID半环中乘法带半环的结构定理,即ID∩.■°D=.■z∨.■z∨D.     

Cyclic semirings with idempotent noncommutative addition

The article discusses the structure of cyclic semirings with noncommutative addition. In the infinite case, the addition is idempotent and is either left or right. Addition of a finite cyclic semirings can be either idempotent or nonidempotent. In the finite additively idempotent cyclic semiring, addition is reduced to the addition of a cyclic subsemiring with commutative addition and an absorbing element for multiplication and the addition of a cycle that is a finite semifield.     

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold Cartesian product of the max-plus semiring: It is known that one can separate a vector from a closed subsemimodule that does not contain it. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 31–52, 2007.     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

The determinability of compacts by lattices of ideals and congruences of semirings of continuous [0, 1]-valued functions on them

We consider an idempotent semiring of continuous [0, 1]-valued functions defined on a compact X with the usual multiplication and addition max. We prove the determinability of X by the lattice of ideals and the lattice of congruences of the indicated semiring.     

Reduced Gröbner Bases of Certain Toric Varieties; A New Short Proof

We prove that every additively-idempotent semiring can be embedded in a finitary complete semiring. From this we obtain, among other results, that the classical identities of Kleene semirings over idempotent semirings are independent.     

On Iteration Semiring-Semimodule Pairs

Conway semiring-module pairs and iteration semiring-semimodule pairs were shown to provide an axiomatic basis to automata on ω -words in [Bloom, Esik: Iteration Theories, Springer, 1993]. In this paper, we show that two natural classes of semiring-semimodule pairs, the complete and the bi-inductive semiring-semimodule pairs both give rise to iteration semiring-semimodule pairs. Complete semiring-semimodule pairs are defined by infinite sums and products, while a bi-inductive semiring-semimodule pair is an ordered semiring-semimodule pair possessing enough least pre-fixed points and greatest post-fixed points to solve linear inequations. Moreover, we show that when V is idempotent, then a semiring-semimodule pair equipped with a star and an omega operation satisfies the Conway equations (iteration semiring-semimodule pair equations, respectively) if and only if the quemiring associated with (S,V) embeds in a Conway semiring (iteration semiring, respectively).     

Solving multi-metric network problems: An interplay between idempotent semiring rules

We motivate computations in a multifunctional networked system as instances of algebraic path problems on labeled graphs. We illustrate, using examples, that composition operators used in many function computations in a networked system follow semiring axioms. We present an abstract framework, using a special idempotent semiring algebraic path problem, to handle multiple metrics for composition. We show that using different vector order relations in this abstract framework, we can obtain different rules of compositions such as Pareto, lexicographic and max-order efficiency. Under this framework, we identify a class of tractable composition rules that can be solved in different multi-criteria settings at affordable computational cost. We demonstrate using an example of trusted routing in which logical security rules of admission control can be combined with delay performance metrics in the multi-criteria optimization framework.     

A class of inherently nonfinitely based semirings

We give a sufficient condition which ensures that a semiring with an idempotent addition is inherently nonfinitely based. This enables us to provide a number of small and natural examples of nonfinitely based semirings, including semirings of binary relations on a finite set. Supported by Grant No.144011 of the Ministry of Science of the Republic of Serbia.     

The structure of idempotent residuated chains

In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains. This work is supported by a grant of NSF, China # 10471112 and a grant of Shaanxi Provincial Natural Science Foundation # 2005A15.