Maximal Elements Research Articles

On Dynamic Aggregation Systems

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭr (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭr (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ2 (A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function fJ,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭr (A) with respect to a set of lots 𝒥, then the aggregation function fJ,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences fJ,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

Reversibility of extreme relational structures

A relational structure $${{\mathbb {X}}}$$ is called reversible iff each bijective homomorphism from $${{\mathbb {X}}}$$ onto $${{\mathbb {X}}}$$ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable $$L_{\infty \omega }$$ -theory defines a class of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs.

A Natural Partial Order on Partition Order-Decreasing Transformation Semigroups

Mitsch defined the natural partial order $$\le $$ on a semigroup S as follows: $$a\le b$$ if and only if $$a=xb=by, a=xa$$ for some $$x,y\in S^1$$ . Let $${{{\mathcal {T}}}}_X$$ be the full transformation semigroup on a finite set $$X=\{1,2,\ldots ,n\}$$ . Let $$\rho $$ be an equivalence relation on X and $$\preceq $$ be a total order on the partition set $$X/\rho $$ of X induced by $$\rho $$ . Denote by $${\overline{x}}$$ the $$\rho $$ -class containing $$x\in X$$ . In this paper, we endow the partition order-decreasing transformation subsemigroup of $${{{\mathcal {T}}}}_X$$ defined by $$\begin{aligned} T(\rho ,\preceq )=\{f\in {{{\mathcal {T}}}}_X: \forall \,\,x\in X, \overline{f(x)}\preceq {\overline{x}}\} \end{aligned}$$ with the natural partial order and give a characterization for this order. Then we determine the compatibility of their elements and find all the minimal and maximal elements.

A nonstable 𝐶*-algebra with an elementary essential composition series

A C ∗ C^* -algebra A \mathcal {A} is said to be stable if it is isomorphic to A ⊗ K ( ℓ 2 ) \mathcal {A} \otimes \mathcal {K}(\ell _2) . Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C ∗ C^* -algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra A \mathcal {A} of B ( ℓ 2 ) \mathcal {B}(\ell _2) , which is the inductive limit of length ω 1 \omega _1 of its separable stable ideals I α \mathcal {I}_\alpha ( α > ω 1 \alpha >\omega _1 ) satisfying I α + 1 / I α ≅ K ( ℓ 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for each α > ω 1 \alpha >\omega _1 , while A \mathcal {A} is not stable. The sequence ( I α ) α ≤ ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} is the GCR composition series of A \mathcal {A} which in this case coincides with the Cantor–Bendixson composition series as a scattered C ∗ C^* -algebra. A \mathcal {A} has the property that all of its proper two-sided ideals are listed as I α \mathcal {I}_\alpha ’s for some α > ω 1 \alpha >\omega _1 , and therefore the family of stable ideals of A \mathcal {A} has no maximal element. By taking A ′ = A ⊗ K ( ℓ 2 ) \mathcal {A}’=\mathcal {A}\otimes \mathcal {K}(\ell _2) we obtain a stable C ∗ C^* -algebra with analogous composition series ( J α ) α > ω 1 (\mathcal {J}_\alpha )_{\alpha >\omega _1} whose ideals J α \mathcal {J}_\alpha are isomorphic to I α \mathcal {I}_\alpha for each α > ω 1 \alpha >\omega _1 . In particular, there are nonisomorphic scattered C ∗ C^* -algebras whose GCR composition series ( I α ) α ≤ ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} satisfy I α + 1 / I α ≅ K ( ℓ 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for all α > ω 1 \alpha >\omega _1 , for which the composition series differs first at α = ω 1 \alpha =\omega _1 .

Topologies for semicontinuous Richter–Peleg multi-utilities

The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition fails to be sufficient. Instead of search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements.

Naturally ordered strong endomorphisms on graphs

In this paper, we study the natural partial order $$\le $$ on SEnd(G), the strong endomorphism monoid of a finite graph G and characterize minimal elements and maximal elements of $$(SEnd(G),\le )$$. Then we introduce the concept of connectedness on $$(SEnd(G),\le )$$ by using the natural partial order $$\le $$ and determine the number of connected components of SEnd(G) under certain conditions.

Optimal Few-Weight Codes From Simplicial Complexes

Recently, some infinite families of binary minimal and optimal linear codes are constructed from simplicial complexes by Hyun {\em et al}. Inspired by their work, we present two new constructions of codes over the ring $\Bbb F_2+u\Bbb F_2$ by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over $\Bbb F_2+u\Bbb F_2$. Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.

Partially ordering the class of invertible trees

A tree T is invertible if and only if T has a perfect matching. In Godsil (1985), Godsil considers an invertible tree T and finds that the matrix A(T)−1 has entries in {0,±1} and is the signed adjacency matrix of a graph which contains T. In this paper, we give a new proof of this theorem, which gives rise to a partial ordering relation on the class of all invertible trees on 2n vertices. In particular, we show that given an invertible tree T whose inverse graph has strictly more edges, we can remove an edge from T and add another edge to obtain a non-isomorphic invertible tree T˜ whose median eigenvalue is strictly greater. This extends naturally to a partial ordering. We find the maximal and minimal elements of this poset and explore the implications about the median eigenvalues of invertible trees.

A-Pseudo complementation on ADL’s

In this paper, we introduce the concept of an [Formula: see text]-pseudo complementation [Formula: see text] on an Almost Distributive Lattice [Formula: see text] for an arbitrary fixed element [Formula: see text] and any [Formula: see text] of [Formula: see text] and discuss several properties of this. Here, we obtain a bijection correspondence between the [Formula: see text]-pseudo complementations on [Formula: see text] and maximal elements of [Formula: see text] We prove that the set [Formula: see text] is a Boolean algebra which is independent(upto isomorphism) of the [Formula: see text]-pseudo complementation [Formula: see text].

Maximal element with applications to Nash equilibrium problems in Hadamard manifolds

ABSTRACTThe purpose of this paper is to study the existence of maximal elements with applications to Nash equilibrium problems for generalized games in Hadamard manifolds. By employing a KKM lemma, we establish a new maximal element theorem in Hadamard manifolds. As applications, some existence results of Nash equilibria for generalized games are derived. The results in this paper unify, improve and extend some known results from the literature.

Exploring feasible design spaces for heterogeneous constraints

We demonstrate an approach of exploring design spaces to simultaneously satisfy kinematics- and physics-based requirements. We present a classification of constraints and solvers to enable postponing optimization as far down the design workflow as possible. The solvers are organized into two broad classes of design space ‘pruning’ and ‘exploration’ by considering the types of constraints they can satisfy. We show that pointwise constraints define feasible design subspaces that can be represented and computed as first-class entities by their maximal feasible elements. The design space is pruned upfront by intersecting maximal elements, without premature optimization. To solve for other constraints, we apply topology optimization (TO), starting from the pruned feasible space. The optimization is steered by a topological sensitivity field (TSF) that measures the global changes in violation of constraints with respect to local topological punctures. The TSF for global objective functions is augmented with TSF for global constraints, and penalized/filtered to incorporate local constraints, including set constraints converted to differentiable (in)equality constraints. We demonstrate application of the proposed workflow to nontrivial examples in design and manufacturing. Among other examples, we show how to explore pruned design spaces via TO to simultaneously satisfy physics-based constraints (e.g., minimize compliance and mass) as well as kinematics-based constraints (e.g., maximize accessibility for machining).

On a partially ordered set associated to ring morphisms

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functor of the category Ring of associative rings with identity to the category ParOrd of partially ordered sets. The maximal elements of Hom(R) constitute a subset Max(R) which, for commutative rings R, can be identified with the Zariski spectrum Spec(R) of R. Every pair (a,M) in Hom(R) has a canonical representative, that is, there is a universal ring morphism ψ:R→S(R/a,M/a) corresponding to the pair (a,M), where the ring S(R/a,M/a) is constructed as a universal inverting R/a-ring in the sense of Cohn. Several properties of the sets Hom(R) and Max(R) are studied.

Deliberately Stochastic

We study stochastic choice as the outcome of deliberate randomization. We derive a general representation of a stochastic choice function where stochasticity allows the agent to achieve from any set the maximal element according to her underlying preferences over lotteries. We show that in this model stochasticity in choice captures complementarity between elements in the set, and thus necessarily implies violations of Regularity/Monotonicity, one of the most common properties of stochastic choice. This feature separates our approach from other models, e.g., Random Utility. (JEL D80, D81)

On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems

Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of L -majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP).

Maximal chains of prime ideals of different lengths in unique factorization domains

We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 \lt n_1 \lt n_2 \lt \cdots \lt n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$ which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.

Determination of pore types in coquinas carbonates and their effects on formation factors and porosity exponents

ABSTRACT Calculation of the saturation of carbonates with various pore types and complicated pore structures in electrical resistivity models requires data on pore types and their effects on electrical resistivity. Therefore, the objectives are to determine the pore types and to investigate their effects on electrical resistivity. The pore types of three coquinas samples were determined by the distributions of experimental porosities and porosity exponents in the cross-plot of the dual porosity model and pore visualization from X-ray computerized tomography. A process-based method was applied to reconstruct the pore-scale models with different pore types. Their pore networks and electrical resistivity were extracted and calculated by the maximal ball and finite element methods, respectively. Fracture reduced the formation factor and porosity exponent. Connected vug increased the pore space that swept by the electrical current to reduce the formation factor while increasing the porosity exponent. Isolated vug undermined the pore space to increase the formation factor and porosity exponent. The effects of the tiny vug were much more obvious than the effects of the large one. This investigation will help to classify the pore types and propose electrical resistivity models that correspond to specific pore types and vug sizes in carbonate reservoirs.

On classy groups of small order

Finite group theory is one of the most delightful areas in mathematics. The ideal aim of finite group theory is to find all finite groups up to isomorphism. In this paper, we reconsider the classification of groups of order less than or equal to 17 by using the presentation techniques of each group with the help of Lagrange’s Theorem. Keywords : Finite groups, Maximal order element, Isomorphism, Generator, Cyclic group, Group presentation

Parameter Optimization of Laminated Multiferroic Composites

The voltage coefficient of a multiferroic laminated composite consisting of magnetostrictive and piezoelectric phases is minimized by optimization of physical parameters of the magnetostrictive material. Two optimization schemes are considered: maximization over a discrete set of values, including parameters of several known materials, and over a continuous domain constrained by the minimal and maximal elements of the discrete set. It is shown that there exist several possible combinations of parameters at which the value of the voltage coefficient is higher than that for known homogeneous materials.

SSB representation of preferences: Weakening of convexity assumptions

A continuous skew-symmetric bilinear (SSB) representation of preferences has recently been proposed in a topological vector space, assuming a weaker notion of convexity of preferences than in the classical (algebraic) case. Equipping a linear vector space with the so-called inductive linear topology, we derive the algebraic SSB representation on such topological basis, thus weakening the convexity assumption. Such a unifying approach to SSB representation leads, moreover, to a stronger existence result for a maximal element and opens a way for a non-probabilistic interpretation of the algebraic theory. Note finally that our method of using powerful topological techniques to derive purely algebraic result may be of general interest.

Fast CS-based pulsar period estimation method without tentative epoch folding and its CRLB

The traditional pulsar period estimation methods have high computational loads as a result of the tentative epoch folding. The tentative epoch folding means that the massive pulsar signals are folded multiple times with different periods. However, due to the low computing power of on-board computer, the tentative epoch folding with high computational load must be avoided. To achieve this objective, we propose a fast compressive sensing-based pulsar period estimation method, which directly estimates the pulsar period according to the distortion degree of the accumulated profile. The compressive sensing (CS) includes three aspects: the dictionary, the measurement matrix and the sparse recovery algorithm. In the proposed method, the pulsar profiles with different distortion degrees comprise the distorted profile dictionary; the low-frequency Fourier transform matrix is developed as the measurement matrix; the super-resolution sparse recovery using the maximal elements is proposed to estimate the pulsar period. Besides, the pulsar period estimation performance against the Cramér-Rao lower bound (CRLB) is studied. And the computational complexity analysis is also provided. The simulation results demonstrate that the fast CS-based pulsar period estimation method approaches the CRLB and its computational time is only 37 ms.