Semigroups Of Relations Research Articles

Ergodicity for time-changed symmetric stable processes

In this paper we study ergodicity and related semigroup property for a class of symmetric Markov jump processes associated with time-changed symmetric α-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric α-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.

Rings with kernel inclusion quasiorder

Building on previous work for semigroups of functions and binary relations, we axiomatize structures consisting of endomorphisms of abelian groups equipped with composition, the usual pointwise operations, and the quasi-order of kernel inclusion. The resulting structures are associative rings enriched by a quasiorder satisfying a finite set of laws. More generally, we axiomatize the kernel inclusion quasi-order on the ring R induced by a right R-module, and we call the resulting abstract structures rings with ker-order. A characterisation of the possible ker-orders on a fixed ring is given in terms of certain families of its right ideals. The lattice of all ker-orders on the ring of rational integers is described.

Right Units in Complete Semigroups of Binary Relations

In this paper, we give a full description of right units of the semigroup B X (D), which are defined by semilattices of the class net. For the case where X is a finite set, we derive formulas by calculating the numbers of right units of the corresponding semigroup.

On moments-preserving cosine families and semigroups in C[0, 1

We use the newly developed Lord Kelvin’s method of images (Bobrowski in J Evol Equ 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1] that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.

Semigroups of Relations and Subdifferentials of the Minimal Time Function and of the Geodesic Distance

Given a closed subset $S$ of a Banach space $X$, we study the minimal time function to reach a point $x$ of $X$ starting from $S$. We relate this problem to the study of the geodesic distance from $x$ to $S$ associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton--Jacobi equation. We detect simple links between the general framework we propose and multivalued semigroups. New concepts of infinitesimal generators of such semigroups are considered.

Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping

This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase spaceD(Δ2)×H01(Ω)∩H2(Ω).

On finitely related semigroups

An algebraic structure is finitely related (has finite degree) if its term func- tions are determined by some finite set of finitary relations. We show that the fol- lowing finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semi- groups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid. This answers a question by Davey, Jackson, Pitkethly, and Szabo from Davey et al. (Semigroup Forum, 83(1):89-122, 2011).

Mitsch’s order and inclusion for binary relations and partitions

Mitsch’s natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described.

Left Cancellative and Right Cancellative Elements of Semigroup of Relations

Let R(X) denote the set of all relations on a set X. Then R(X) becomes a semigroup under composition. The purpose of this paper is to describe the element of R(X) which is left cancellative and right cancellative.

Property of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions

We study properties of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions.

Complete semigroups of binary relations defined by finite XI-semilattices of unions

We give conditions under which the X-semilattice of unions is an XI-semilattice, i.e., let D be a finite X-semilattice of unions. The complete semigroup of binary relation BX(D) is defined by the XI-semilattice of unions if and only if V (D, β) = D for some β ∈ BX(D).

From Diffusions on Graphs to Markov Chains via Asymptotic State Lumping

We show that fast diffusions on finite graphs with semi permeable membranes on vertices may be approximated by finite-state Markov chains provided the related permeability coefficients are appropriately small. The convergence theorem involves a singular perturbation with singularity in both operator and boundary/transmission conditions, and the related semigroups of operators converge in an irregular manner. The result is motivated by recent models of synaptic depression.

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n − 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n −1(F).

Complete semigroups of binary relations defined by X-semilattices of unions

This paper is devoted to some results from the book “Complete Semigroups of Binary Relations,” which is not published yet. Conditions of divisibility of binary relations, idempotents, regular elements, one-side units, and one-side zeros are described. Finite X-semilattices and unions are characterized. Methods of finding formulas for counting the number of idempotents and regular elements are given.

Long‐Term Damped Dynamics of the Extensible Suspension Bridge

This work is focused on the doubly nonlinear equation , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness k2. When the ends are pinned, long‐term dynamics is scrutinized for arbitrary values of axial load p and stiffness k2. For a general external source f, we prove the existence of bounded absorbing sets. When f is time‐independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B σ (X) in the case where σ(X) is a quasi-order.

Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations

We study linear differential operators with unbounded operator-valued coefficients acting on homogeneous spaces of functions on the semi-axis. We obtain necessary and sufficient conditions for such operators to be invertible or Fredholm. Essential use is made of the spectral theory of difference relations and semigroups of linear relations (multi-valued linear operators).

Global attractors for the extensible thermoelastic beam system

This work is focused on the dissipative system { ∂ t t u + ∂ x x x x u + ∂ x x θ − ( β + ‖ ∂ x u ‖ L 2 ( 0 , 1 ) 2 ) ∂ x x u = f , ∂ t θ − ∂ x x θ − ∂ x x t u = g describing the dynamics of an extensible thermoelastic beam, where the dissipation is entirely contributed by the second equation ruling the evolution of θ. Under natural boundary conditions, we prove the existence of bounded absorbing sets. When the external sources f and g are time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity for all parameters β ∈ R . The same result holds true when the first equation is replaced by ∂ t t u − γ ∂ x x t t u + ∂ x x x x u + ∂ x x θ − ( β + ‖ ∂ x u ‖ L 2 ( 0 , 1 ) 2 ) ∂ x x u = f with γ > 0 . In both cases, the solutions on the attractor are strong solutions.

Regular elements of semigroups B X (D) defined by generalized elementary B X (D)-semilattices of unions

In the paper, the class of complete semigroups of binary relations is considered, each of whose elements is defined by a complete BX(D)-semilattice of unions which belongs to the class of generalized elementary X-semilattices. Regular elements are described for each semigroup of this class.

Idempotents and regular elements of complete semigroups of binary relations

As is known (see [5, 6]), idempotent elements of semigroups play an important role in the investigation of semigroups themselves. Moreover, it is known that any semigroup can be isomorphically embedded into some semigroup of binary relations on some nonempty set X. This has generated great interest in the investigation of idempotent binary relations. Different authors used different approaches to study the construction of these elements and obtained different answers to the question under consideration (see, e.g., [1–3, 7, 8]). In this work, first, we characterize the complete X-semilattices of unions D for which there exist idempotent binary relations such that the set of all their cuts to the elements of D coincides with the given semilattice. Then we give a description of the structure of these idempotent binary relations in the language of limiting elements of some subset of the semilattice D (see Lemma 1 and Theorem 5) and show how we can find all idempotents of the complete group of binary relations BX (D) defined by the complete X-semilattice of unions D (see Theorem 6 and Corollary 4). Moreover, with the help of chains of the semilattice D, we give a rule for constructing the semigroups of idempotent elements of the semigroup BX(D) (see Theorem 7). In Theorem 9, we give a description of the structure of all regular elements of the semigroup BX (D).