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.     

几类加法非正则半环

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.     

On isotone functions with the Substitution Property in distributive lattices

A universal algebra isaffine complete if all functions satisfying the Substitution Property are polynomials (composed of the basic operations and the elements of the algebra). In 1962, the first author proved that a bounded distributive lattice is affine complete if and only if it does not contain a proper Boolean interval. Recently, M. Ploica generalized this result to arbitrary distributive lattices.In this paper, we introduce a class of functions on a latticeL, we call themID-polynomials, that derive from polynomials on the ideal lattice (resp., dual ideal lattice) ofL; they are isotone functions and satisfy the Substitution Property. We prove that for a distributive latticeL, all unary functions with the Substitution Property are ID-polynomials if and only ifL contains no proper Boolean interval.The research of the first author was supported by the NSERC of Canada. The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Congruence lattices of function lattices

Thefunction lattice L ^P is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL ^P is a direct power of the congruence lattice ofL; the exponent is |P|.This result fails for infiniteP. However, utilizing a generalization of theL ^P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.In this paper we prove that the isomorphism ConL[D](ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

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.     

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.     

A Characterization of Semirings Which Are Subdirect Products of a Distributive Lattice and a Ring

E -inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.     

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.     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

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.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

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.     

Remarks on the lattices of fuzzy subsets of a groupoid

In this paper we give a necessary and sufficient condition for a groupoid D such that the sup-min product is distributive over arbitrary intersection of fuzzy subsets of D, and correct some results from the paper [S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239–245]. Also, we prove that the set of all idempotent fuzzy sets forms a complete lattice, which is a complete join-sublattice of the lattice of all fuzzy subgroupoids. This result extends the corresponding result from the above mentioned paper.     

Combinatorial partitions of finite posets and lattices —Ramsey lattices

It is proved that for every finite latticeL there exists a finite latticeL such that for every partition of the points ofL into two classes there exists a lattice embeddingf:LL such that the points off(L) are in one of the classes.This property is called point-Ramsey property of the class of all finite lattices. In fact a stronger theorem is proved which implies the following: for everyn there exists a finite latticeL such that the Hasse-diagram (=covering relation) has chromatic number >n. We discuss the validity of Ramseytype theorems in the classes of finite posets (where a full discussion is given) and finite distributive lattices. Finally we prove theorems which deal with partitions of lattices into an unbounded number of classes.Presented by G. Grätzer.     

Ein Distributivitätskriterium für Verbände

Those one-place functions on a latticeL with values inL which can be obtained from the identityx and the constant functions by means of the fundamental operations ofL in a finite number of steps are called one-place polynomial functions. —It is shown that a latticeL is distributive iff all its one-place polynomial functions can be represented by words such that in every word the indeterminatex occurs at most once.     

An algorithm for determining Φ(L) in finite distributive lattices

Following G. Birkhoff, the Frattini sublattice Φ(L) of a latticeL is defined as the intersection of all its maximal proper sublattices. Let ?(FD) be the class of all finite distributive lattices. The main aim in this note is to provide a new but elementary characterization of elements in Φ(L),L∈?(FD), and also an extremely simple algorithm for determining the Frattini sublattice of any finite distributive lattice. By applying this algorithm, it is shown that there is a new way to determine the Frattini subalgebra of a finite Stone algebra.     

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.     

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

For compact Hausdorff Lawson lattices L that are idempotent semirings with lower semicontinuous multiplications, free compact Hausdorff Lawson L-idempotent semimodules are constructed over compacta and over compact Hausdorff Lawson lower semilattices. It is shown that these free objects are spaces of L-valued regular measures. Related left adjoint functors and monads are described.     

Lattices of quotients of completely distributive lattices

In this paper, we consider the complete lattice Q(L) of all quotients of a completely distributive lattice L. We show that Q(L) is not a completely distributive lattice even for L a completely distributive algebraic lattice. Some necessary and sufficient conditions for Q(L) to be a completely distributive lattice are given. Received February 26, 2003; accepted in final form January 17, 2005.