Prime Ideals in Birational Extensions of Two-Dimensional Power Series Rings

In this article we consider simple birational extensions of power series rings in one variable over one-dimensional Noetherian domains having infinitely many maximal ideals. For these rings we describe the partially ordered sets that arise as prime spectra. We characterize the prime spectra in the case that the coefficient rings are countable Dedekind domains. The prime spectra over Dedekind domains are the same as the prime spectra that arise for simple birational extensions of power series rings over the integers and the same as the prime spectra of simple birational extensions of k[[x]][z], where k is a countable field and x and z are indeterminates.     

Trivializable rings of sets

In the present paper, we examine the ways in which rings of sets can be generated by as simple systems as possible. We investigate rings of sets which can be generated by totally ordered subsystems. All totally ordered systems which generate the same ring are, in essence, isomorphic. This fact leads to an order-theoretic classification of rings of sets. Every group-valued mapping defined on a totally ordered system of sets can be uniquely extended to an additive mapping on the ring of sets generated by the system. Two rings of sets are of the same order type if and only if the corresponding spaces of measures are isomorphic.     

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ? T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.     

Comparability invariance of the fixed point property

We show that the fixed point property is comparability invariant for finite ordered sets; that is, if P and Q are finite ordered sets with isomorphic comparability graphs, then P has the fixed point property if and only if Q does. In the process we give a characterization of comparability invariants which can also be used to give shorter proofs of some known results.     

Finitely presented and coherent ordered modules and rings

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features:.

(i) Every subring of Q, and every totally ordered division ring, is po-coherent.

(ii) For a partially ordered right module Aover a po-coherent poring R Ais po-coherent if and only if Ais a finitely presented .R-module and A ⁺is a finitely generated R ⁺-semimodule.

(iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent.

(iv) Every finitely po-presented abelian lattice-ordered group is po-coherent.     

Algorithms over partially ordered sets

We here study some problems concerned with the computational analysis of finite partially ordered sets. We begin (in § 1) by showing that the matrix representation of a binary relationR may always be taken in triangular form ifR is a partial ordering. We consider (in § 2) the chain structure in partially ordered sets, answer the combinatorial question of how many maximal chains might exist in a partially ordered set withn elements, and we give an algorithm for enumerating all maximal chains. We give (in § 3) algorithms which decide whether a partially ordered set is a (lower or upper) semi-lattice, and whether a lattice has distributive, modular, and Boolean properties. Finally (in § 4) we give Algol realizations of the various algorithms.     

阿基米德序半群的链(英文)

本文通过一个序半群S上的一些二元关系以及它的理想(右理想,双理想)的根集分别给出了该序半群是阿基米德(右阿基米德,t-阿基米德)序子半群的链的刻画.进一步证明了准素序半群是阿基米德序半群的链.最后,通过素根定理证明了序半群S是阿基米德序子半群的链当且仅当S是阿基米德序子半群的半格且S的所有素理想关于集合的包含关系构成链.     

Totally ordered sets and the prime spectra of rings

Let T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain O_v such that T is order isomorphic to two special subsets of Spec(O_v). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U?Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.     

Affine and projective lines over one-dimensional semilocal domains

We characterize those partially ordered sets that can occur as the spectra of polynomial rings over one-dimensional semilocal (Noetherian) domains. We also determine the posets that can occur as projective lines over one-dimensional semilocal domains.

     

Homomorphism-Homogeneous Partially Ordered Sets

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between partially ordered sets A and B is a mapping f : A →B satisfying ). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X ₅-dense locally bounded partially ordered sets. Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017.     

On Ordered Sets with Isomorphic Marked Maximal Cards

We prove a necessary condition for two nonisomorphic ordered sets to have two isomorphic marked maximal cards. This condition is used to prove that ordered sets of width 3 with 2 maximal elements are reconstructible and that ordered sets of width 3 are reconstructible if we can reconstruct the marked maximal deck.     

On countable completions of quotient ordered semigroups

A poset is said to be ω-chain complete if every countable chain in it has a least upper bound. It is known that every partially ordered set has a natural ω-completion. In this paper we study the ω-completion of partially ordered semigroups, and the topological action of such a semigroup on its ω-completion. We show that, for partially ordered semigroups, ω-completion and quotient with respect to congruences are two operations that commute with each other. This contrasts with the case of general partially ordered sets.     

ASSOCIATIONS AND VALUATIONS

Abstract

Right cones are semigroups for which the lattice of right ideals is a chain and a left cancellation law holds; valuation rings, the cones of ordered groups, and initial segments of ordinal numbers are examples. Two such cones are associated if they have isoniorphic lattices of right ideals so that ideals, prime ideals, and completely prime ideals correspond to each other. A list of problems is discussed. In Proposition 3.11 it is proved that the canonical mapping from a right invariant right chain domain R onto the associated right holoid can be extended to a valuation from the skew field Q(R) of quotients of R onto an ordered group if and only if Ja ? aJ for all a ∈ R and J = J(R), the Jacobson radical of R.     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximal dimensional partially ordered sets II. characterization of 2n-element posets with dimension n

In this paper, we show that if a partially ordered set has 2n elements and has dimension n, then it is isomorphic to the set of n?1 element subsets and 1element subsets of a set, ordered by inclusion, or else it has six elements and is isomorphic to a partially ordered set we call the chevron or to its dual.     

有限級有序加羣及有序環的表現

在另一文中,我们讨论了由全体2维實向量所成的有序环,在该文最後並说當维数n>2时(n为有限)也可类似地作初步讨论.为了显示这种向量环的用途,我们考虑用向量环来表现一般有序环的问题.在本文中我们证明:任一“n级的”(见以下定义)有序环都能与一个由若干n维實向量所组成的有序环同构.(主要在於证出关於n级有序加羣的类似结果.)我们希望有较好的结果,即:任一n级有序环都能与由全体n维實向量所成的一个有序环的一个子环同构,但未能证明或否定.     

The convexity lattice of a poset

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.     

Interval orders based on weak orders

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “ ”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “ ”, a six-element crown or a six-element fence.     

Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

阿基米德序半群的链的若干刻画

本文首先引入了一个序半群$S$的准素模糊理想的概念,通过序半群$S$上的一些二元关系以及它的理想的模糊根给出了该序半群是阿基米德序子半群的半格的一些刻画.进一步地借助于序半群$S$的模糊子集对该序半群是阿基米德序子半群的半格进行了刻画.尤其是通过序半群的模糊素根定理证明了序半群$S$是阿基米德序子半群的链当且仅当$S$是阿基米德序子半群的半格且$S$的所有弱完全素模糊理想关于模糊集的包含关系构成链.