A topological characterization of the goldman prime spectrum of a commutative ring

A prime ideal p of a commutative ring R is said to be a Goldman ideal (or a G-ideal) if there exists a maximal ideal M of the polynomial ring R[X] such that p = M ∩ R. A topological space is said to be goldspectral if it is homeomorphic to the space Gold(R) of G-ideals of R (Gold(R) is considered as a subspace of the prime spectrum Spec(R) equipped with the Zariski topology). We give here a topological characterization of goldspectral spaces.     

A prime Baer group

Mathematische Zeitschrift -     

A topological completeness theorem

We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. Received: 27 September 1996 / Revised version: 15 January 1998     

A procedure to compute prime filtration

Let K be a field, S = K[x ₁, … x _n ] be a polynomial ring in n variables over K and I ⊂ S be an ideal. We give a procedure to compute a prime filtration of S/I. We proceed as in the classical case by constructing an ascending chain of ideals of S starting from I and ending at S. The procedure of this paper is developed and has been implemented in the computer algebra system Singular.     

A local prime factor decomposition algorithm

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived.Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known “classical” prime factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products.     

A topological representation of lattices

Presented by R. S. Pierce.     

A topological notion of boundedness

A topological boundedness notion is studied, which is proved to be productive. Classical theorems on compactness of Tychonoff, Alexander and Obreanu are generalized. A boundedness operator is defined and studied. Finally, a classification of all topological spaces is obtained according to boundedness criteria.The author is grateful to prof. N. Oeconomidis, who suggested the topic, for his continuous interest.     

A topological colorful Helly theorem

Let be d+1 families of convex sets in . The Colorful Helly Theorem (see (Discrete Math. 40 (1982) 141)) asserts that if for all choices of then there exists an 1?i?d+1 such that .Our main result is both a topological and a matroidal extension of the colorful Helly theorem. A simplicial complex X is d-Leray if for all induced subcomplexes Y⊂X and i?d.Theorem.LetXbe ad-Leray complex on the vertex setV. Suppose M is a matroidal complex on the same vertex setVwith rank functionρ. IfM⊂Xthen there exists a simplexτ∈Xsuch thatρ(V−τ)?d.     

A Boolean topological orthomodular poset

Wilce introduced the notion of a topological orthomodular poset and proved any compact topological orthomodular poset whose underlying orthomodular poset is a Boolean algebra is a topological Boolean algebra in the usual sense. Wilce asked whether the compactness assumption was necessary for this result. We provide an example to show the compactness assumption is necessary.     

A topological central point theorem

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.     

A topological approach to evasiveness

The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the graph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity Ω(v ²) wherev is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper the truth of Karp’s conjecture is shown to follow from another conjecture concerning group actions on topological spaces. A special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices. Supported in part by an NSF postdoctoral fellowship Supported in part by NSF under grant No. MCS-8102248