A note on almost maximal chain rings

2020 ◽  
Vol 144 ◽  
pp. 1-11
Author(s):  
Ulrich Albrecht ◽  
Francisco Javier Santillán-Covarrubias
Keyword(s):  
Maximal Chain

scholarly journals Traces Without Maximal Chains

10.37236/465 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ta Sheng Tan
Keyword(s):  
Maximal Chain ◽  
Maximal Chains

The trace of a family of sets ${\cal A}$ on a set $X$ is ${\cal A}|_X=\{A\cap X:A\in {\cal A}\}$. If ${\cal A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace ${\cal A}|_X$ does not contain a maximal chain, then how large can ${\cal A}$ be? Patkós conjectured that, for $n$ sufficiently large, the size of ${\cal A}$ is at most ${n-k+r-1\choose r-1}$. Our aim in this paper is to prove this conjecture.

Binary Trees and the n-Cutset Property

1991 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Peter Arpin ◽  
John Ginsburg
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Binary Tree ◽  
Ordered Set ◽  
Maximal Chain ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximal Elements ◽  
Extremal Case ◽  
Result Theorem

AbstractA partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n. Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω. If P — Max P is countable then P contains a complete binary tree of height ω

scholarly journals A maximal chain approach to topology and order

1988 ◽  
Vol 11 (3) ◽  
pp. 465-472 ◽  
Author(s):  
R. Vainio
Keyword(s):  
Ordered Set ◽  
Maximal Chain ◽  
Ordered Sets ◽  
Complete Lattices ◽  
Interval Topology ◽  
Convergence Structure ◽  
Order Structures ◽  
General Convergence

On ordered sets (posets, lattices) we regard topologies (or, more general convergence structures) which on any maximal chain of the ordered set induce its own interval topology. This construction generalizes several well-known intrinsic structures, and still contains enough to produce interesting results on for instance compactness and connectedness. The “maximal chain compatibility” between topology (convergence structure) and order is preserved by formation of arbitrary products, at least in case the involved order structures are conditionally complete lattices.

A Proof of the Maximal Chain Theorem

1952 ◽  
Vol 74 (3) ◽  
pp. 676 ◽  
Author(s):  
Orrin Frink
Keyword(s):  
Maximal Chain

Notes on Local Integral Extension Domains

1978 ◽  
Vol 30 (01) ◽  
pp. 95-101 ◽  
Author(s):  
L. J. Ratliff
Keyword(s):  
Prime Ideal ◽  
Maximal Chain ◽  
Integral Closure ◽  
Prime Ideals ◽  
Local Domain ◽  
Important Paper ◽  
Integral Extension ◽  
Local Integral ◽  
Maximal Ideals ◽  
Extension Domain

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3]. In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

scholarly journals Maximal chain continuous factor

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Noriaki Kawaguchi
Keyword(s):  
Maximal Chain

A special chain theorem in the set of intermediate rings

2017 ◽  
Vol 16 (10) ◽  
pp. 1750185 ◽  
Author(s):  
Mabrouk Ben Nasr ◽  
Nabil Zeidi
Keyword(s):  
Maximal Chain ◽  
Main Tool ◽  
Integral Closure ◽  
Explicit Description ◽  
Integral Domains ◽  
Intermediate Ring

Let [Formula: see text] be an extension of integral domains, and let [Formula: see text] be the integral closure of [Formula: see text] in [Formula: see text]. The main purpose of this paper is to study [Formula: see text], the set of intermediate rings between [Formula: see text] and [Formula: see text]. As a main tool, we establish an explicit description of any intermediate ring in terms of localizations of [Formula: see text] (or [Formula: see text]). This study effectively enables us to characterize the minimal extensions in [Formula: see text]. We also prove a special chain theorem concerning the length of an arbitrary maximal chain in [Formula: see text].

scholarly journals Chains of congruences on a completely 0-simple semigroup

1965 ◽  
Vol 5 (1) ◽  
pp. 76-82 ◽  
Author(s):  
G. B. Preston
Keyword(s):  
Finite Length ◽  
Simple Semigroup ◽  
Maximal Chain

Let ρ and σ be two congruences on a completely 0-simple semigroup. Suppose that there is a maximal chain of congruences from ρ to σ which is of finite length. Then, as we shall show, any maximal chain of congruences from ρ to σ finite and of the same length.

On the Number of Maximal Elements in a Partially Ordered Set

1987 ◽  
Vol 30 (3) ◽  
pp. 351-357 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Maximal Chain ◽  
Maximal Elements ◽  
Partially Ordered

AbstractLet P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

A maximal chain of principal ideals in the semigroup of binary relations on a finite set

1991 ◽  
Vol 43 (1) ◽  
pp. 63-76 ◽  
Author(s):  
Michael Breen
Keyword(s):  
Maximal Chain ◽  
Binary Relations ◽  
Principal Ideals ◽  
Finite Set
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