scholarly journals Congruence Class Sizes in Finite Sectionally Complemented Lattices

2004 ◽  
Vol 47 (2) ◽  
pp. 191-205 ◽  
Author(s):  
G. Grätzer ◽  
E. T. Schmidt
Keyword(s):  
Congruence Class ◽  
Typical Result ◽  
Complemented Lattices ◽  
The Family ◽  
Complemented Lattice ◽  
Congruence Classes

AbstractThe congruences of a finite sectionally complemented lattice L are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence Θ of L is from being uniform, we introduce Spec Θ, the spectrum of Θ, the family of cardinalities of the congruence classes of Θ. A typical result of this paper characterizes the spectrum S = (mj | j < n) of a nontrivial congruence Θ with the following two properties:

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals ACDC: Analysis of Congruent Diversification Classes

2022 ◽  
Author(s):  
Sebastian Hoehna ◽  
Bjoern Tore Kopperud ◽  
Andrew F Magee
Keyword(s):  
Congruence Class ◽  
R Package ◽  
Rate Function ◽  
Diversification Rate ◽  
Extinction Rate ◽  
Diversification Rates ◽  
Common Features ◽  
Alternative Hypotheses ◽  
Rate Functions ◽  
Congruence Classes

Diversification rates inferred from phylogenies are not identifiable. There are infinitely many combinations of speciation and extinction rate functions that have the exact same likelihood score for a given phylogeny, building a congruence class. The specific shape and characteristics of such congruence classes have not yet been studied. Whether speciation and extinction rate functions within a congruence class share common features is also not known. Instead of striving to make the diversification rates identifiable, we can embrace their inherent non-identifiable nature. We use two different approaches to explore a congruence class: (i) testing of specific alternative hypotheses, and (ii) randomly sampling alternative rate function within the congruence class. Our methods are implemented in the open-source R package ACDC (https://github.com/afmagee/ACDC). ACDC provides a flexible approach to explore the congruence class and provides summaries of rate functions within a congruence class. The summaries can highlight common trends, i.e. increasing, flat or decreasing rates. Although there are infinitely many equally likely diversification rate functions, these can share common features. ACDC can be used to assess if diversification rate patterns are robust despite non-identifiability. In our example, we clearly identify three phases of diversification rate changes that are common among all models in the congruence class. Thus, congruence classes are not necessarily a problem for studying historical patterns of biodiversity from phylogenies.

Families of even non-congruent numbers with prime factors in each odd congruence class modulo eight

2018 ◽  
Vol 14 (03) ◽  
pp. 669-692
Author(s):  
Lindsey Reinholz ◽  
Blair K. Spearman ◽  
Qiduan Yang
Keyword(s):  
Congruence Class ◽  
Prime Factors ◽  
Congruence Classes

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight.

Equational Classes of Distributive Pseudo-Complemented Lattices

1970 ◽  
Vol 22 (4) ◽  
pp. 881-891 ◽  
Author(s):  
K. B. Lee
Keyword(s):  
Unary Operation ◽  
Complemented Lattices ◽  
Complemented Lattice

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.

The lattices of families of regular sets in topological spaces

2020 ◽  
Vol 70 (2) ◽  
pp. 477-488
Author(s):  
Emilia Przemska
Keyword(s):  
Closure Operator ◽  
Boolean Algebras ◽  
Topological Spaces ◽  
Closed Sets ◽  
The Family ◽  
Interior Operator ◽  
Regular Sets ◽  
Regular Open ◽  
Complemented Lattice ◽  
Regular Closed

Abstract The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

How many maximal surfaces do correspond to one minimal surface?

2009 ◽  
Vol 146 (1) ◽  
pp. 165-175 ◽  
Author(s):  
HENRIQUE ARAÜJO ◽  
MARIA LUIZA LEITE
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Hausdorff Space ◽  
Quotient Space ◽  
Closed Interval ◽  
Congruence Class ◽  
Quotient Topology ◽  
Maximal Surfaces ◽  
Hopf Differential ◽  
Congruence Classes

AbstractWe discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

scholarly journals On semigroups whose nontrlvial left congruence classes are left ideals

1971 ◽  
Vol 12 (1) ◽  
pp. 1-5
Author(s):  
E. Hotzel
Keyword(s):  
Equivalence Class ◽  
Left Ideal ◽  
Equivalence Relation ◽  
Pairwise Disjoint ◽  
Congruence Class ◽  
Identity Relation ◽  
Class I ◽  
Left Congruence ◽  
Congruence Classes

An equivalence relation δ on a semigroup S is called a left congruence of S if (u, v) Ε λ implies that (su, sv) Ε λ for every s in S. With every set ℒ of pairwise disjoint left ideals (i. e. subsets L of S such that SL⊂EL), one can associate the left congruence {(u, v)|u = v or there exists an L in ℒ such that u Ε L and v Ε L}. Thus every nonempty left ideal is a left congruence class (i. e. an equivalence class of some left congruence). A left congruence has the form just described if and only if all its nontrivial classes (i. e. its classes containing at least two elements) are left ideals. Such a left congruence is called a Rees left congruence if there is at most one nontrivial class. The identity relation on S is a Rees left congruence since the empty set is a left ideal by definition.

scholarly journals Guaranteed Scoring Games

10.37236/5417 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Urban Larsson ◽  
Richard J. Nowakowski ◽  
João P. Neto ◽  
Carlos P. Santos
Keyword(s):  
Congruence Class ◽  
Order Relation ◽  
Combinatorial Games ◽  
Partially Ordered ◽  
Ordered Monoid ◽  
Partially Ordered Monoid ◽  
Congruence Classes

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by 'switching the players'. The order relation between two games is defined by comparing their stops in any disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

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