scholarly journals On the endomorphism semigroup of an ordered set

1995 ◽  
Vol 37 (2) ◽  
pp. 173-178 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Maximal Element ◽  
Ordered Set ◽  
Minimal Element ◽  
Hasse Diagram ◽  
Endomorphism Semigroup ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Image Position

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of(a) all antichains;(b) all quasi-complete chains;(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);(f) the six-element crown C6 with Hasse diagramA similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

scholarly journals A Theorem on Partially Ordered Sets and its Application

1969 ◽  
Vol 9 (3-4) ◽  
pp. 361-362
Author(s):  
Vladimir Devidé
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Image Position

Let (S, ≦) be a (non-void) partially ordered set with the property that for every (non-void) chain C (i.e., every totally ordered subset) of S, there exists in S the element sup C. Let SM be the set of all maximal elements s of S. ƒ:S/SM→S be a slowly increasing mapping in the sense that

LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 463-482 ◽  
Author(s):  
EKATERINA FOKINA ◽  
BAKHADYR KHOUSSAINOV ◽  
PAVEL SEMUKHIN ◽  
DANIEL TURETSKY
Keyword(s):  
Ordered Set ◽  
Equivalence Relations ◽  
Ordered Sets ◽  
Linear Orders ◽  
Algebraic Properties ◽  
Image Position ◽  
Quotient Set ◽  
The Relationship ◽  
Induced Structure ◽  
Linearly Ordered Set

AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

scholarly journals Groups of automorphisms of linearly ordered sets

1976 ◽  
Vol 15 (1) ◽  
pp. 13-32 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Simple Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Ordered Set ◽  
Embedding Theorems ◽  
Ordered Sets ◽  
Infinite Cyclic Group ◽  
Groups Of Automorphisms ◽  
Linearly Ordered Set

I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdös-Hajnal inductive classification of scattered sets.

scholarly journals NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

2012 ◽  
Vol 87 (1) ◽  
pp. 94-107 ◽  
Author(s):  
LEI SUN ◽  
LIMIN WANG
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Nonempty Subset ◽  
Transformation Semigroup ◽  
Invariant Set ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Minimal Elements ◽  
Image Position

AbstractLet 𝒯X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

scholarly journals Ordered Structures of Constructing Operators for Generalized Riesz Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Hiroshi Inoue
Keyword(s):  
Hilbert Space ◽  
Ordered Set ◽  
Closed Operator ◽  
Inner Product ◽  
Maximal Elements ◽  
Ordered Structures ◽  
Minimal Elements ◽  
Riesz System ◽  
Densely Defined

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.

ON RAMSEY’S THEOREM AND THE EXISTENCE OF INFINITE CHAINS OR INFINITE ANTI-CHAINS IN INFINITE POSETS

2016 ◽  
Vol 81 (1) ◽  
pp. 384-394 ◽  
Author(s):  
ELEFTHERIOS TACHTSIS
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Infinite Chain ◽  
Ordered Sets ◽  
Ramsey's Theorem ◽  
Partially Ordered ◽  
Ramsey’S Theorem ◽  
Image Position ◽  
The Axiom Of Choice

AbstractRamsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle.In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel–Mostowski permutation model ${\cal N}$ for ZFA and prove that the above principle for infinite partially ordered sets is true in ${\cal N}$, whereas Ramsey’s Theorem is false in ${\cal N}$. Then, based on the existence of ${\cal N}$ and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem.In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true in Mostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.

scholarly journals THE EXPECTED JAGGEDNESS OF ORDER IDEALS

2017 ◽  
Vol 5 ◽  
Author(s):  
MELODY CHAN ◽  
SHAHRZAD HADDADAN ◽  
SAM HOPKINS ◽  
LUCA MOCI
Keyword(s):  
Probability Distribution ◽  
Young Diagram ◽  
Partially Ordered Sets ◽  
Order Ideal ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Combinatorial Theorem ◽  
Order Ideals ◽  
Geometric Formula

Thejaggednessof an order ideal$I$in a poset$P$is the number of maximal elements in$I$plus the number of minimal elements of$P$not in$I$. A probability distribution on the set of order ideals of$P$istoggle-symmetricif for every$p\in P$, the probability that$p$is maximal in$I$equals the probability that$p$is minimal not in$I$. In this paper, we prove a formula for the expected jaggedness of an order ideal of $P$under any toggle-symmetric probability distribution when$P$is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015,arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

scholarly journals Galois and Pataki Connections on Generalized Ordered Sets

2019 ◽  
pp. 283-323
Author(s):  
Árpád Száz
Keyword(s):  
Present Author ◽  
Ordered Set ◽  
Ordered Sets

In this paper, having in mind Galois and Pataki connections, we establish several basic theorems on increasingly seminormal and semiregular functions between gosets. An ordered pair X(\leq )=(X,\leq ) consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set). A function f of one goset X to another Y is called increasingly upper g-seminormal, for some function g of Y to X, if f(x)\leq y implies x \leq g(y). While, the function f is called increasingly upper φ-semiregular, for some function φ of X to itself, if f(u)\leq f(v) implies u\leq \varphi (v). The increasingly lower seminormal (semiregular) functions are defined by the reverse implications. Moreover, a function is called increasingly normal (regular) if it is both increasingly upper and lower seminormal (semiregular). The results obtained extend and supplement several former results of O. Ore and the present author on Galois and Pataki connections. Namely, the pairs (f, g) and (f, φ) may be called increasing Galois and Pataki connections if the function f is increasingly g-normal and φ-regular, respectively.

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