Adjacency on the order polytope with applications to the theory of fuzzy measures

In this paper we study the adjacency structure of the order polytope of a poset. For a given poset, we determine whether two vertices in the corresponding order polytope are adjacent. This is done through filters in the original poset. We also prove that checking adjacency between two vertices can be done in quadratic time on the number of elements of the poset. As particular cases of order polytopes, we recover the adjacency structure of the set of fuzzy measures and obtain it for the set of p-symmetric measures for a given indifference partition; moreover, we show that the set of p-symmetric measures can be seen as the order polytope of a quotient set of the poset leading to fuzzy measures. From this property, we obtain the diameter of the set of p-symmetric measures. Finally, considering the set of p-symmetric measures as the order polytope of a direct product of chains, we obtain some other properties of these measures, as bounds on the volume and the number of vertices on certain cases.     

Uniserial modules: sums and isomorphisms of subquotients

Let R be an associative ring with 1. A left R module M is uniserial i f the lattice L(M) of its submodules is totally ordered under inclusion. We give an example of a uniserial module M with the property of having two submodules 0 < H < K < M such that M is isomorphic to K/H (we call a module M with this property shrinkable). Then we give an example of a uniserial module M isomorphic to all its nonzero quotients M/N, N<M, and with L(M) isomorphic to ω²+1; this solves a problem of Hirano and Mogami [7]. Finally we show that for uniserial modules the property of being shrinkable is connected to the problem of deciding whether a module, which is both a homomorphic image of a finite direct sum of uniserial modules and a submodule of a finite direct sum of uniserial modules, is a finite direct sum of uniserial modules     

Decomposing torsion modules

E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R_ and let K be the cokernel of the canonical ring homomorphism from R to R_. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R,  and K, there is a related filter _K of ideals of R, which is a subset of , such that all _K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between  and _K is investigated.     

Prime, irreducible elements and coatoms in posets

In this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.     

$QTAG$-Modules whose $h$-Pure-$S$-High Submodules Have Closure

A right module $M$ over an associative ring $R$ with unity is a $QTAG$-module if every finitely generated submodule of any homomorphic image of $M$ is a direct sum of uniserial modules. This article considers the closure of $h$-pure-$S$-high submodules of $QTAG$-modules. Here, we determine all submodules $S$ of a $QTAG$-module $M$ such that each closure of $h$-pure-$S$-high submodule of $M$ is $h$-pure-$\overline{S}$-high in $\overline{M}$. A few results of this theme give a comparison of some elementary properties of $h$-pure-$S$-high and $S$-high submodules.     

A note on the structure of injective diagrams

We determine all indecomposable injective diagrams over a poset I with values in some category R-Mod of modules and characterize the case when every injective diagram decomposes into a direct sum of indecomposables. Moreover, we show that injective diagrams have a standard form dual to that of projective diagrams iff I is a noetherian poset.     

Decomposition of injective modules relative to a torsion theory

IfR is a right noetherian ring, the decomposition of an injective module, as a direct sum of uniform submodules, is well known. Also, this property characterises this kind of ring. M. L. Teply obtains this result for torsion-free injective modules. The decomposition of injective modules relative to a torsion theory has been studied by S. Mohamed, S. Singh, K. Masaike and T. Horigone. In this paper our aim is to determine those rings satisfying that every torsion-freeτ-injective module is a direct sum ofτ-uniformτ-injective submodules and also to determine those rings with the same property for everyτ-injective module.     

Cyclically decomposable distributive modules

Let R be a ring and M an R-module. Then M is said to be distributive if the lattice of submodules of M is distributive. We determine the structure of distributive modules, and show that in certain cases a distributive module is either cyclic or is a direct sum of cyclic submodules.     

可图序列偏序集的极小元

ｎ项非增非负整数序列是可图的,若是某个阶简单图的度序列．所有项和为２ｍ、迹为ｆ的ｎ项可图序列的集合Ｇ_ｎ,ｍ,ｆ在优超关系下是一个偏序集．本文刻划了偏序集Ｇ_ｎ,ｍ,ｆ的极小元,并确定各种可图序列偏序集中极小元的个数．     

Poset extensions, convex sets, and semilattice presentations

The paper is devoted to an algebraic and geometric study of the feasible set of a poset, the set of finite probability distributions on the elements of the poset whose weights satisfy the order relationships specified by the poset. For a general poset, this feasible set is a barycentric algebra. The feasible sets of the order structures on a given finite set are precisely the convex unions of the primary simplices, the facets of the first barycentric subdivision of the simplex spanned by the elements of the set. As another fragment of a potential complete duality theory for barycentric algebras, a duality is established between order-preserving mappings and embeddings of feasible sets. In particular, the primary simplices constituting the feasible set of a given finite poset are the feasible sets of the linear extensions of the poset. A finite poset is connected if and only if its barycentre is an extreme point of its feasible set. The feasible set of a (general) disconnected poset is the join of the feasible sets of its components. The extreme points of the feasible set of a finite poset are specified in terms of the disjointly irreducible elements of the semilattice presented by the poset. Semilattices presented by posets are characterised in terms of various distributivity concepts.     

The lumpability property for a family of Markov chains on poset block structures

We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We essentially use combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associated with a Gelfand pair, the spectral analysis of the corresponding lumped Markov chain is completely determined by the decomposition of the group action into irreducible submodules.     

The Möbius function of permutations with an indecomposable lower bound

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.     

Going down in (semi)lattices of finite moore families and convex geometries

In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then we study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set—ordered by set inclusion—is a ranked join-semilattice and we characterize its cover relation. We prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.      

Poset Loops

Given a ring and a locally finite poset, an incidence loop or poset loop is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.     

蕴含权重的偏序集多准则决策法

针对偏序集方法不能解决含有权重的多准则决策问题,提出一种“隐式”赋权的偏序决策方法。首先将含有m个方案和n个准则的决策问题表示成偏序集,之后按权重由大到小的顺序,对准则进行逐步相加形成n个新的准则,得到一个新的偏序集。根据偏序集间的包含关系,证明了新偏序集不仅蕴含了权重信息,而且比初始偏序集有更强的排序能力。结果表明,该法在应用中仅需获取权重排序信息,无需精确权重,适用于权重难以确定的多准则决策问题。以三峡库区水质评价为例,例子表明新方法明显优于原有的偏序决策方法,能够对13个方案进行聚类和排序,而原有方法在该例中几乎难以应用。     

Partial differential equation approach to E 7

By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E₇ into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E₇. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan's well-known fourth-order E₇-invariant partial differential equation.     

A partial order for the set of meanders

In this paper, we present a new approach for studying meanders in terms of noncrossing partitions. We show how this approach leads to a natural partial order on the set of meanders. In particular, meanders form a graded poset with regard to this partial order.     

Trees and circle orders

This paper continues a recent resurgence of interest in combinatorial properties of a poset that are associated with graph properties of its cover graph and order diagram. The following two theorems appearing in a 1977 paper of Trotter and Moore have played important roles in motivating this more modern research: (1) The dimension of a poset is at most 3 when its cover graph is at tree; (2) The dimension of a poset is at most 3 when the poset has a zero and its order diagram is planar. Although the underlying ideas lay dormant for more than 30 years, the first of these two results has become the base case for recent results bounding the dimension of a poset in terms of (a) the tree-width of its cover graph, and (b) the maximum dimension of its blocks. The second result is the base case for bounding the dimension of a planar poset in terms of the number of minimal elements. Continuing with this line of research, we show that every poset whose cover graph is a tree is a circle order, i.e., it has a representation as a family of circular disks in the Euclidean plane partially ordered by inclusion.     

Semigroup actions on posets and preimage quasi-orders

Structures consisting of a semigroup of (partial) functions on a set X, a?poset of subsets of X, and a preimage operation linking the two, arise commonly throughout mathematics. The poset may be equipped with one or more set operations, up to Boolean algebra structure. Such structures are finitely axiomatized here in terms of order-preserving semigroup actions on posets. This generalises Schein??s axiomatization of semigroups of partial functions equipped with the first projection quasi-order.     

分层有限偏序集的直积

A new concept Graded Finite Poset is proposed in this paper.Through discussing some basic properties of it,we come to that the direct product of graded finite posets is connected if and only if every graded finite poset is connected.The graded function of a graded finite poset is unique if and only if the graded finite poset is connected.