Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k‐homogeneous if any isomorphism between k‐element subsets extends to an automorphism of (P, ≤). Assuming the set‐theoretic assumption ⋄(ϰ₁), it is shown that for each k, there exist partially ordered sets of size ϰ₁ which embed each countable partial order and are k‐homogeneous, but not (k + 1)‐homogeneous. This is impossible in the countable case for k ≥ 4.     

Cycle Orders

Let X, T and C be, respectively, a finite set with at least three points, a set of ordered triples of distinct points from X, and a cyclic ordering of the points in X. Define T C to mean that, for every a b c T, the elements a, b, c occur in that cyclic order in C, and let C(T) denote the set of cyclic orderings of X for which T C. We say that T is noncyclic if C(T) is empty, cyclic if C(T) is nonempty, uniquely cyclic if | C(T) | = 1, a partial cycle order if it is cyclic and T ={a b c :{a b c} C for all C C(T)}, and a total cycle order if it is a uniquely cyclic partial cycle order. Many years ago E. V. Huntington axiomatized total cycle orders by independent necessary and sufficient conditions on T. The present paper studies the more relaxed structures of cyclic T sets and partial cycle orders. We focus on conditions for cyclicity, a theory of cycle dimension of partial cycle orders, and extremal problems that address combinatorial structures of T sets.     

Linear Extensions and Comparable Pairs in Partial Orders

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on n elements, which has close to a third of the pairs comparable with high probability: we show that the number of linear extensions is n! 2^?Θ(n) with high probability.     

Additive Maps Preserving the Star Partial Order on B(H)

Let B(H) be the C*-algebra of all bounded linear operators on a complex Hilbert space H. It is proved that an additive surjective map φ on B(H) preserving the star partial order in both directions if and only if one of the following assertions holds.(1) There exist a nonzero complex number α and two unitary operators U and V on H such that φ(X) = αUXV or φ(X) = αUX*V for all X ∈ B(H).(2)There exist a nonzero α and two anti-unitary operators U and V on H such thatφ(X) = αUXV or φ(X) = αUX*V for all X ∈ B(H).     

Linear Extensions of Semiorders: II

Let M _n (k) denote the family of posets on n points with k ordered pairs that maximize the number of linear extensions among all such posets. Fishburn and Trotter [2] prove that every poset in M _n (k) is a semiorder and identifies all semiorders in M _n (k) for k n. The present paper specifies M _n (k) for all k 2 n – 3.     

富足半群上的自然偏序

本文研究富足半群上的自然偏序，得到Green关系和自然偏序之间的联系，确定了富足半群何时关于自然偏序具有单边(双边)相容，另外，也研究了富足半群的本原元。     

交换偏序半群的偏序扩张

设(S,·,≤)为偏序可换半群,本文给出将S的偏序≤扩张为满足一定条件的偏序≤^{*的充要条件.特别地,如果(S,·,≤)为可消偏序可换幺半群,本文给出将S的偏序≤扩张为可消偏序≤*且S的每个元素在≤*下均在正锥中的充要条件.本文还给出将偏序可换幺半群S的偏序≤扩张为≤*且使得S的有限元素子集在≤*下是一条链的充要条件.}     

A Characterization and the Intersection of the Maximal Compatible Extensions of a Partial Order

We present a characterization of the maximal compatible extensions of a given compatible partial order ≤  _r on a unary algebra (A,f ). These extensions can be constructed by using the compatible linear extensions of ≤  _r*, where (A*,f*) is the so called contracted quotient algebra of (A,f) and the compatible partial order ≤  _r* on (A*,f*) is naturally induced by ≤  _r . Using this characterization, we determine the intersection of the maximal compatible extensions of ≤  _r .      

Elementary Properties of Cycle-free Partial Orders and their Automorphism Groups

A classification was given by Creed, Truss, and Warren of all the countable k-CS-transitive cycle-free partial orders for k3. Here the elementary theories of these structures and their automorphism groups are examined, and it is shown that in many cases we can distinguish the structures or their groups by means of their first- or second-order properties. The small index property is established for weakly 2-transitive trees, and for several classes of cycle-free partial orders.     

右逆半群上的偏序关系

It is well known that there exists the smallest inverse semigroup congruence on an orthodox semigroup. We denote by Y the smallest inverse semigroup congruence on an orthodox semigroup. Let S be a fight inverse semigroup. We construct partial orders on S by some kind of its subsemigroups and uncover that partial orders on S have close contact with partial orders on S/Y.     

Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$

Let $\mathcal{B}(\mathcal{H})$ be the $C^∗$-algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. It is proved that an additive surjective map $φ$ on $\mathcal{B}(\mathcal{H})$ preserving the star partial order in both directions if and only if one of the following assertions holds. (1) There exist a nonzero complex number $α$ and two unitary operators $\boldsymbol{U}$and$\boldsymbol{V}$ on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$. (2) There exist a nonzero $α$ and two anti-unitary operators$\boldsymbol{U}$and$\boldsymbol{V}$on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$.     

算子偏序的刻画及性质

本文定义了Hilbert空间上两个算子间的四种关系:星序、左星序,右星序及减序,使用了算子分块矩阵的方法,给出了两个算子具有上述四种关系之一时它们几何结构的刻画,证明了这四种关系是真正的偏序关系,进一步研究了它们之间的关系和性质.     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

On k-CS-Transitive Cycle-Free Partial Orders with Finite Alternating Chains

The notion of cycle-free partial order (CFPO) was defined by R. Warren, and the major cases of the classification of the countable sufficiently transitive CFPOs were given, the finite and infinite chain cases, by Creed, Truss, and Warren. It is the purpose of this paper to complete the classification. The cases which remained untreated were CFPOs not embedding an infinite alternating chain ALT (which can only happen in the finite chain case). It is shown that if a k-CS-transitive CFPO does not embed ALT, where k 3, then it does not embed any alternating chain of size k + 3, and this leads to the desired classification (which is only given explicitly for k = 3 and 4). The general result says that the class of k-CS-transitive CFPOs for k 3 not embedding ALT admits a recursive classification.     

Extending Partial Orders on o-Minimal Structures to Definable Total Orders

It is shown that if (M, <, ?) is an o-minimal structure such that (M, <) is a dense total order and ? is a parameter-definable partial order on M, then ? has an extension to a definable total order.     

Certain Partial Orders on Semigroups

Relations introduced by Conrad, Drazin, Hartwig, Mitsch and Nambooripad are discussed on general, regular, completely semisimple and completely regular semigroups. Special properties of these relations as well as possible coincidence of some of them are investigated in some detail. The properties considered are mainly those of being a partial order or compatibility with multiplication. Coincidences of some of these relations are studied mainly on regular and completely regular semigroups.     

Antichains and Linear Extensions

Order - When {1, 2,..., m} is an antichain in a poset on m + n points, how should the other n points be arranged to maximize the proportion of linear extensions in which 1 &;gt; 2 &;gt;...     

Cube Orders

A finite partially ordered set P =(X, ) is a cube order in R ^m if there is a map C from X into cubes in R ^m such that, for all x and y in X, x y if and only if C(x) C(y). It is proved that every finite poset with order dimension 2 m is a cube order in R ^m .     

Partial Orders on Weak Orders Convex Subsets

We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.