The ideal-theory of the partially ordered set

1941 ◽  
Vol 13 (2) ◽  
pp. 94-99 ◽  
Author(s):  
R. Vaidyanathaswamy
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Ideal Theory ◽  
Partially Ordered ◽  
The Ideal

scholarly journals Rings which resemble rings of entire functions

1983 ◽  
Vol 24 (1) ◽  
pp. 7-16
Author(s):  
David E. Rush
Keyword(s):  
Riemann Surface ◽  
Partially Ordered Set ◽  
Riemann Surfaces ◽  
Entire Functions ◽  
Ordered Set ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Topological Spaces ◽  
Valuation Theory ◽  
The Ideal

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

The space of formal balls and models of quasi-metric spaces

2009 ◽  
Vol 19 (2) ◽  
pp. 337-355 ◽  
Author(s):  
M. ALI-AKBARI ◽  
B. HONARI ◽  
M. POURMAHDIAN ◽  
M. M. REZAII
Keyword(s):  
Metric Space ◽  
Partially Ordered Set ◽  
Metric Spaces ◽  
Ordered Set ◽  
Space Problem ◽  
Partially Ordered ◽  
Sorgenfrey Line ◽  
Maximal Point ◽  
Ideal Completion ◽  
The Ideal

In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, ⊑, ≺). We prove that if the conjugate space (X, d−1) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, ⊑, ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

scholarly journals Each join-completion of a partially ordered set in the solution of a universal problem

1974 ◽  
Vol 17 (4) ◽  
pp. 406-413 ◽  
Author(s):  
Jürgen Schmidt
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Unique Extension ◽  
Partially Ordered ◽  
Special Cases

The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

ON RIBBON PRESENTATIONS OF RIBBON KNOTS

1994 ◽  
Vol 03 (02) ◽  
pp. 223-231
Author(s):  
TOMOYUKI YASUDA
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Totally Ordered Set

A ribbon n-knot Kn is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kn has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓp AND ℓq

2015 ◽  
Vol 80 (3) ◽  
pp. 917-939 ◽  
Author(s):  
ZHI YIN
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Partially Ordered

AbstractWe prove that, for 1 ≤ p < q < ∞, the partially ordered set P(ω)/Fin can be embedded into Borel equivalence relations between ℝω/ℓp and ℝω/ℓq. Since there is an antichain of size continuum in P(ω)/Fin, there are continuum many pairwise incomparable Borel equivalence relations between ℝω/ℓp and ℝω/ℓq.

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