Maximal Elements Research Articles

Multilattices on typical hesitant fuzzy sets

In this paper, we mainly study multilattices on the set of finite nonempty subsets of [0,1] under different orders. Firstly, the relationships among orders ≤ 1, ≤ 2 and ≤ * are discussed. It is shown that the order ≤ * is contained in the orders ≤ 1 and ≤ 2, but the converse is not true, and that there is no inclusion between the orders ≤ 1 and ≤ 2. The lattice and multilattice structures of the set of finite nonempty subsets of [0,1] under ≤ 1 and ≤ * are studied, respectively. It is proved that the set of finite nonempty subsets of [0,1] is neither a lattice nor a multilattice with respect to the order ≤ *, and is not a multilattice with respect to the order ≤ 1. Finally, we extend order ≤ * to the set V([0, 1]) of finite vectors of [0,1] and develop a new order ⪯. It is verified that the quotient set V([0, 1])/ ≡ under the order ⪯ is not a lattice, but is a multilattice. Moreover, we give a method to construct maximal elements and minimal elements for finite vectors of [0,1] with respect to the order ⪯. The results will provide an application of hesitant fuzzy sets in rough sets, information fusion, fuzzy automata, fuzzy transitive systems, etc.

The maximum of the 1-measurement of a metric measure space

For a metric measure space, we consider the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have the Lipschitz order relation introduced by M. Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement of a metric measure space with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.

Natural partial order and finiteness conditions on semigroups of linear transformations with invariant subspaces

Given a vector space V and a subspace W of V, we consider the semigroup (under composition) S(V, W), consisting of all linear transformations on V which leave the subspace W invariant. In this paper, we characterize the natural partial order on S(V, W). In the partially ordered set, we determine the compatibility of their elements, and find all maximal and minimal elements. Also, we give necessary and sufficient conditions for S(V, W) to be factorizable, unit-regular and directly finite.

The Jones-Wenzl idempotent of a generalized Temperley-Lieb algebra

We define an idempotent of the Temperley-Lieb algebra of any finite Coxeter system, which reduces, in type A, to the known Jones-Wenzl idempotent. We give, for any pair of parabolic subgroups, a recursive formula generalizing the well-known one. This approach gives a wide class of recursive formulas for the classical Jones-Wenzl idempotent. We also compute explicitly the coefficient corresponding to the maximal element of any minuscule quotient, when the idempotent is expressed in the basis of fully commutative elements.

Upper Semicontinuous Representability of Maximal Elements for Nontransitive Preferences

We characterize the possibility of determining all the maximal elements for a preorder on a topological space by maximizing all the functions in a suitable family of upper semicontinuous order-preserving functions. We discuss the possibility of extending this result to the case of a quasi-preorder (i.e., a reflexive and Suzumura consistent binary relation) on a topological space.

An inversion statistic for reduced words

We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a statistic on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this statistic extends naturally to balanced labellings, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.

On the genus of the k-maximal hypergraph of commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text], a fixed integer. Let [Formula: see text] be the set of all [Formula: see text]-maximal elements in [Formula: see text] Associate a [Formula: see text]-maximal hypergraph [Formula: see text] to [Formula: see text] with vertex set [Formula: see text] and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we determine all isomorphism classes of finite commutative non-local rings with identity whose [Formula: see text]-maximal hypergraph has genus one. Finally, we classify all finite commutative non-local rings [Formula: see text] for which [Formula: see text] is projective.

A note on assignment games with the same nucleolus

We show that the family of assignment matrices which give rise to the same nucleolus forms a compact join-semilattice with one maximal element. The above family is, in general, not a convex set, but path-connected.

Minimizing the Difference of Dual Functions of Two Coradiant Functions

In this article, we solve the problem of minimizing the difference of dual functions of two coradiant functions. We do this by applying a type of duality, that is used in microeconomic theory. Indeed, the dual function of a co-radiant function is decreasing and inverse coradiant. So, we first give various characterizations for the maximal elements of the support sets of this class of functions. Next, by using these results, we obtain the necessary and sufficient conditions for the global minimizers of the difference of two decreasing and inverse coradiant functions. Finally, as an application, we present the necessary and sufficient conditions for the global minimizers of the difference of dual functions of two co-radiant functions.

Embeddability on functions: Order and chaos

We study the quasi-order of topological embeddability on definable functions between Polish 0 0 -dimensional spaces. We consider the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order. We also investigate the existence of maximal elements with respect to embeddability in a given Baire class. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

Generalized Birkhoff Center of Almost Distributive Lattices

In this paper we define sectional Birkhoff center for an Almost Distributive Lattice not necessarily with maximal elements; as a Birkhoff center of its principal ideals. Moreover we extend this result and define the generalized Birkhoff center of an ADL not necessarily with maximal elements. We give a necessary and sufficient condition for an ADL to be relatively complemented in terms of its sectional Birkhoff centers. Also we define and characterize sectionaly complemented ideals and sectional factor congruences using sectional Birkhoff centers. MSC-2010: 06D99 Key words: Birkhoff center of ADLs; Sectional Birkhoff center of ADLs; generalized Birkhoff center of ADLs.

Finding Maximal Independent Elements of Products of Partial Orders (The Case of Chains)

We consider one of the central intractable problems of logical data analysis – finding maximal independent elements of partial orders (dualization of a product of partial orders). An important particular case is considered with each order a chain. If each chain is of cardinality 2, the problem involves the construction of a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form. An asymptotic expression is obtained for a typical number of maximal independent elements of products for a large number of finite chains. The derivation of such asymptotic bounds is a technically complex problem, but it is necessary for the proof of existence of asymptotically optimal algorithms for the monotone dualization problem and the generalization of this problem to chains of higher cardinality. An asymptotically optimal algorithm is described for the problem of dualization of a product of finite chains.

Characterizing approximate global minimizers of the difference of two abstract convex functions with applications

In this paper, we first investigate characterizations of maximal elements of abstract convex functions under a mild condition. Also, we give various characterizations for global "-minimum of the difference of two abstract convex functions and, by using the abstract Rockafellar?s antiderivative, we present the abstract ?-subdifferential of abstract convex functions in terms of their abstract subdifferential. Finally, as an application, a necessary and sufficient condition for global ?-minimum of the difference of two increasing and positively homogeneous (IPH) functions is presented.

Central Sets Theorem on noncommutative semigroups

Let D be a directed set without maximal element, S be an infinite semigroup and DS be the collection of all functions from D into S. It is shown that for a commutative semigroup S, A⊆S is a C-set with respect to NS if and only if A is a C-set with respect to DS. We investigate the Central Sets Theorem for arbitrary semigroups. In fact the Central Sets Theorem is stated with respect to SS for arbitrary semigroups.

A Variational Principle for Ground Spaces

The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice elements and the maximal lattice elements within the set of all subspaces using constraints on operator cones. Our results contribute to the geometry of quantum marginals, as their lattices of exposed faces are isomorphic to the lattices of ground spaces of local Hamiltonians.

Ordered Structures of Constructing Operators for Generalized Riesz Systems

A sequence {φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.

The minimal and maximal symmetries for J-contractive projections

In this paper, we first characterize the structure of symmetries J such that a projection P is J-contractive. Then the minimal and maximal elements of the symmetries J with P⁎JP⩽J (or JP⩾0) are given. Moreover, some formulas between P(2I−P−P⁎)+(P(2I−P−P⁎)−) and P(P+P⁎)−(P(P+P⁎)+) are established.

A note on “linear optimization with bipolar max-min constraints” [Information Sciences 234 (2013) 3–15

The conditions (36) for existence of the solution in the decomposed constraints (35) in [2] are not necessary and sufficient. These conditions are only necessary conditions. Also, the authors in [2] claimed that the maximal elements of the system of bipolar max-min equality constraints or set D are either g+ or belong to the set G={Supi=1,...,mgi|(∀i)(gi∈Gi−)}. This expression is not correct. The set G is revised such that it contains the maximal elements of D.

Some Generalizations of Fixed Point Theorems of Caristi Type and Mizoguchi–Takahashi Type Under Relaxed Conditions

In this paper, we first study the approximate fixed point property for hybrid Caristi type and Mizoguchi–Takahashi type mappings on metric spaces. We give some new generalizations of Mizoguchi–Takahashi’s fixed point theorem and Caristi’s fixed point theorem under new relaxed conditions which are quite original in the existing literature. We present new generalized Ekeland’s variational principle, generalized Takahashi’s nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. Their equivalence relationships are also established.

Subalgebra lattices of totally reflexive sub-preprimal algebras

For a set Q of relations on a finite set A, the set $$\mathrm{Pol}_{A}Q$$ , of all operations on A preserving all relations in Q, is a clone. The set of all clones on a given set forms a lattice under inclusion. Each of its maximal elements can be represented as $$\mathrm{Pol}_{A}\{\rho \}$$ where $$\rho $$ is one of the six types of relations determined by Ivo G. Rosenberg. Four types of them are totally reflexive. These relations are bounded orders, non-trivial equivalence relations, central relations, and universal relations. An algebra $$\underline{A}$$ is said to be totally reflexive sub-preprimal if its clone of term operations is where $$\rho _{1}$$ and $$\rho _{2}$$ are totally reflexive relations. We describe the subalgebra lattices of such algebras.