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.     

Varieties Generated by Ordered Bands I

Ordered bands are regarded as semirings whose multiplicative reduct is a band and whose additive reduct is a chain. We find the variety of semirings generated by all ordered bands and we determine part of the lattice of its subvarieties.     

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based. Project supported by the National Natural Science Foundation of China (Grant No. 19761004) and the Provincial Applied Fundamental Research Foundation of Yunnan (96a001z).     

Distributive lattice decompositions of semirings with a semilattice additive reduct

We describe the least distributive lattice congruence on the semirings in the variety of all semirings whose additive reduct is a semilattice, introduce the notion of a k-Archimedean semiring and characterize the semirings that are distributive lattices or chains of k-Archimedean semirings.     

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     

On semirings whose additive reduct is a semilattice

A semiring S whose additive reduct is a semilattice is called a k-regular semiring if for every a∈S there is x∈S such that a+axa=axa. For a semigroup F, the power semiring P(F) is a k-regular semiring if and only if F is a regular semigroup. An element e∈S is a k-idempotent if e+e ²=e ². Basic properties of k-regular semirings whose k-idempotents are commutative have been studied.     

Congruence openings of additive Green’s relations on a semiring

In a series of papers, Green’s relations on the additive and multiplicative reducts of a semiring proved to be a very useful tool in the study of semirings. However, in the vast majority of cases, Green’s relations are not congruences, and we show that in such cases it is much more convenient to use the congruence openings of Green’s relations, instead of the Green’s relations themselves. By means of these congruence openings we define and study several very interesting operators on the lattices of varieties of semirings and additively idempotent semirings, and, in particular, we establish order embeddings of the lattice of varieties of additively idempotent semirings into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice that we introduced.     

On the subsemiring generated by almost idempotents of a k-regular semiring

An element e of a semiring S with commutative addition is called an almost idempotent if \(e + e^2 = e^2\). Here we characterize the subsemiring \(\langle E(S)\rangle \) generated by the set E(S) of all almost idempotents of a k-regular semiring S with a semilattice additive reduct. If S is a k-regular semiring then \(\langle E(S)\rangle \) is also k-regular. A similar result holds for the completely k-regular semirings, too.     

A commutativity theorem

LetR be an associative ring with identity which satisfies the identities(xy) ^k =(yx) ^k and(xy) ^l =(yx) ^l , wherek andl are relatively prime positive integers, depending onx andy. ThenR is commutative. Moreover, examples are given which show thatR need not be commutative if either of the above identities is dropped. This theorem is also true for groups. Presented by N. S. Mendelsohn.     

Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions

In this work, lattice isomorphisms of semirings C ⁺(X) of continuous nonnegative functions over an arbitrary topological space X are characterized. It is proved that any isomorphism of lattices of all subalgebras with a unit of semirings C ⁺(X) and C ⁺(Y) is induced by a unique isomorphism of semirings. The same result is also correct for lattices of all subalgebras excepting the case of two-point Tychonovization of spaces.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

The varieties of semilattice-ordered semigroups satisfying $$x^3\approx x$$ and $$xy\approx yx$$xy≈yx

The aim of this paper is to study the varieties of semilattice-ordered Burnside semigroups satisfying \(x^3\approx x\) and \(xy\approx yx.\) It is shown that the collection of all such varieties forms a distributive lattice of order 9. Also, all of them are finitely based and finitely generated. This gives a generalization and expansion of the results obtained by McKenzie and Romanowska (Contrib Gen Algebra Proc Klagenf Conf 1978 1:213–218, 1979).     

Finitely additive,modular, and probability functions on pre-Semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the law of total probability, Bayes’ theorem, the equality of parallel systems, and Poincaré’s inclusion-exclusion theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,?), where D is a Dedekind domain.     

E-special Ordered Regular Semigroups

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x ⁺ ∈ S such that both xx ⁺ and x ⁺ x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x ⁺ is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S ⁺ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.     

Associativity of the <Emphasis Type="Italic">∇</Emphasis>-operation on bands in rings

Given a multiplicative band of idempotents S in a ring R, for all e,f∈S the ∇-product e ∇ f=e+f+fe−efe−fef is an idempotent that lies roughly above e and f in R just as ef and fe lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.     

Congruence-simple subsemirings of ℚ+

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class CongSimp(\mathbb Q⁺)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of CongSimp(\mathbbQ⁺)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ⁺)\{\mathbbQ⁺}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.     

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.     

Varieties of idempotent slim groupoids

Idempotent slim groupoids are groupoids satisfying xx ≈ x and x(yz) ≈ xz. We prove that the variety of idempotent slim groupoids has uncountably many subvarieties. We find a four-element, inherently nonfinitely based idempotent slim groupoid; the variety generated by this groupoid has only finitely many subvarieties. We investigate free objects in some varieties of idempotent slim groupoids determined by permutational equations. The work is a part of the research project MSM0021620839 financed by MSMT and partly supported by the Grant Agency of the Czech Republic, grant #201/05/0002.     

Standard Bases of a Vector Space Over a Linearly Ordered Incline

Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.