A minimal prime model with an infinite set of indiscernibles

1972 ◽  
Vol 11 (2) ◽  
pp. 180-183 ◽  
Author(s):  
Leo Marcus
Keyword(s):  
Prime Model ◽  
Minimal Prime ◽  
Infinite Set

A Note on Minimal Prime Divisors of an Ideal

2011 ◽  
Vol 18 (spec01) ◽  
pp. 727-732
Author(s):  
Kamal Bahmanpour ◽  
Ahmad Khojali ◽  
Reza Naghipour
Keyword(s):  
Integral Domain ◽  
Quotient Ring ◽  
Regular Sequence ◽  
Prime Divisors ◽  
Normal Ring ◽  
Ring Of Polynomials ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Infinite Set ◽  
Equivalent Conditions

We explore several equivalent conditions for finiteness of the set of minimal prime divisors of an ideal and conclude the results of Anderson, Gilmer and Heinzer as especial cases. It is proved that in the ring of polynomials K[X], in which K is a Noetherian ring and X a (possibly infinite) set of indeterminates over K, these conditions are necessary and sufficient. In particular, it is proved that in K[X], an ideal has a finite number of minimal prime divisors if and only if all its minimal prime divisors have finite height. The same results are proved for the derived normal ring of a Noetherian integral domain and the quotient ring K[X]/I, in which I is generated by a K[X]-regular sequence of finite length. We also give a counterexample to show that the conditions of Anderson, Gilmer and Heinzer are not sufficient.

scholarly journals Homomorphisms of distributive p-algebras with countably many minimal prime ideals

1987 ◽  
Vol 35 (3) ◽  
pp. 427-439 ◽  
Author(s):  
M. E. Adams ◽  
V. Koubek ◽  
J. Sichler
Keyword(s):  
Boolean Algebras ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Zero Semigroup ◽  
Categorical Properties ◽  
The Right ◽  
Minimal Prime ◽  
Countably Infinite ◽  
Infinite Set

According to a result of Lee, varieties of pseudocomplemented distributive lattices form an ω+1 chain in which is the trivial variety and is the variety of Boolean algebras. In the present paper it is shown that the variety contains an almost universal subcategory B in which the members of Hom(B,B') associated with minimal prime ideals of B form a countably infinite set for any B,B' ∈ B. In particular, B3contains arbitrarily large algebras whose nontrivial endomorphisms form the countably infinite right zero semigroup. Our earlier results concerning categorical properties of varieties of pseudocomplemented distributive lattices show that no further reduction of the right zero count is possible.

Number of countable models

1978 ◽  
Vol 43 (3) ◽  
pp. 492-496 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Complete Theory ◽  
Prime Model ◽  
Categorical Theory ◽  
Definable Subset ◽  
Algebraic Elements ◽  
Minimal Prime ◽  
Countable Theory ◽  
The Universe ◽  
Countable Models

We prove that a countable complete theory whose prime model has an infinite definable subset, all of whose elements are named, has at least four countable models up to isomorphism. The motivation for this is the conjecture that a countable theory with a minimal model has infinitely many countable models. In this connection we first prove that a minimal prime model A has an expansion by a finite number of constants A′ such that the set of algebraic elements of A′ contains an infinite definable subset.We note that our main conjecture strengthens the Baldwin–Lachlan theorem. We also note that due to Vaught's result that a countable theory cannot have exactly two countable models, the weakest possible nontrivial result for a non-ℵ0-categorical theory is that it has at least four countable models.§1. Notation and preliminaries. Our notation follows Chang and Keisler [1], except that we denote models by A, B, etc. We use the same symbol to refer to the universe of a model. Models we refer to are always in a countable language. For T a countable complete theory we let n(T) be the number of countable models of T up to isomorphism. ∃n means ‘there are exactly n’.

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

An Infinite Set of Heron Triangles with Two Rational Medians

1997 ◽  
Vol 104 (2) ◽  
pp. 107-115 ◽  
Author(s):  
Ralph H. Buchholz ◽  
Randall L. Rathbun
Keyword(s):  
Infinite Set

scholarly journals Time and Life in the Relational Universe: Prolegomena to an Integral Paradigm of Natural Philosophy

Philosophies ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 30 ◽  
Author(s):  
Abir Igamberdiev
Keyword(s):  
Probability Function ◽  
Natural Philosophy ◽  
Natural World ◽  
Spatiotemporal Structure ◽  
Internal Logic ◽  
Decision Making System ◽  
Relational Domains ◽  
Relational Order ◽  
Infinite Set ◽  
The Universe

Relational ideas for our description of the natural world can be traced to the concept of Anaxagoras on the multiplicity of basic particles, later called “homoiomeroi” by Aristotle, that constitute the Universe and have the same nature as the whole world. Leibniz viewed the Universe as an infinite set of embodied logical essences called monads, which possess inner view, compute their own programs and perform mathematical transformations of their qualities, independently of all other monads. In this paradigm, space appears as a relational order of co-existences and time as a relational order of sequences. The relational paradigm was recognized in physics as a dependence of the spatiotemporal structure and its actualization on the observer. In the foundations of mathematics, the basic logical principles are united with the basic geometrical principles that are generic to the unfolding of internal logic. These principles appear as universal topological structures (“geometric atoms”) shaping the world. The decision-making system performs internal quantum reduction which is described by external observers via the probability function. In biology, individual systems operate as separate relational domains. The wave function superposition is restricted within a single domain and does not expand outside it, which corresponds to the statement of Leibniz that “monads have no windows”.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

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