Extending Algebras to Model Congruence Schemes

1986 ◽  
Vol 38 (2) ◽  
pp. 257-276 ◽  
Author(s):  
J. Berman ◽  
G. Grätzer ◽  
C. R. Platt
Keyword(s):  
Universal Algebra ◽  
Congruence Relation ◽  
Principal Congruence ◽  
Algebraic Functions ◽  
Congruence Relations ◽  
Image Position

This paper is concerned with the description of principal congruence relations. Given elements a and b of a universal algebra , let θ(a, b) denote the smallest congruence relation on containing the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says that c ≡ d(θ(a, b)) if and only if there exists a sequence z0, z1, …, zn of elements of and a sequence f1, f2, …, fn of unary algebraic functions such that c = z0, d = zn, and for each i = 1, …, n,Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice of a, b, c and d. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

Congruence Relations Between the Traces of Matrix Powers

1949 ◽  
Vol 1 (3) ◽  
pp. 303-304 ◽  
Author(s):  
J. S Frame
Keyword(s):  
Finite Order ◽  
Rational Integer ◽  
Roots Of Unity ◽  
Finite Degree ◽  
Characteristic Roots ◽  
Congruence Relations ◽  
Image Position

Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain nth roots of unity a1, a2…, ad. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruenceeven though tr(Ap) may not itself be rational.

scholarly journals Extensional quotient coalgebras

2017 ◽  
Vol 9 (2) ◽  
pp. 303-323
Author(s):  
Jean-Paul Mavoungou
Keyword(s):  
Congruence Relation ◽  
Maximal Element ◽  
Congruence Relations

Abstract Given an endofunctor F of an arbitrary category, any maximal element of the lattice of congruence relations on an F-coalgebra (A, a) is called a coatomic congruence relation on (A, a). Besides, a coatomic congruence relation K is said to be factor split if the canonical homomor-phism ν : AK → A∇A splits, where ∇A is the largest congruence relation on (A, a). Assuming that F is a covarietor which preserves regular monos, we prove under suitable assumptions on the underlying category that, every quotient coalgebra can be made extensional by taking the regular quotient of an F-coalgebra with respect to a coatomic and not factor split congruence relation or its largest congruence relation.

scholarly journals Local polynomial functions on factor rings of the integers

1979 ◽  
Vol 27 (2) ◽  
pp. 232-238 ◽  
Author(s):  
Hans Lausch ◽  
Wilfried Nöbauer
Keyword(s):  
Abelian Group ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Commutative Rings ◽  
Infinite Length ◽  
Polynomial Functions ◽  
Local Polynomial ◽  
Definition Of ◽  
Image Position

AbstractLet A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.

scholarly journals Congruence relations of Ankeny-Artin-Chowla type for pure cubic fields

1984 ◽  
Vol 96 ◽  
pp. 95-112 ◽  
Author(s):  
Hiroshi Ito
Keyword(s):  
Elliptic Curve ◽  
Class Number ◽  
Congruence Relation ◽  
Bernoulli Numbers ◽  
Quadratic Fields ◽  
Hurwitz Numbers ◽  
Real Quadratic Fields ◽  
Congruence Relations ◽  
Cubic Fields ◽  
Real Quadratic

Ankeny, Artin and Chowla [1] proved a congruence relation among the class number, the fundamental unit of real quadratic fields, and the Bernoulli numbers. Our aim of this paper is to prove similar congruence relations for pure cubic fields. For this purpose, we use the Hurwitz numbers associated with the elliptic curve defined by y2 = 4x3 — 1 instead of the Bernoulli numbers (§ 3).

Free Involutorial Completely Simple Semigroups

1985 ◽  
Vol 37 (2) ◽  
pp. 271-295 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Maximal Subgroup ◽  
Universal Algebra ◽  
Simple Semigroup ◽  
Binary Operation ◽  
Unary Operation ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Image Position ◽  
Completely Simple

An involution x → x* of a semigroup S is an antiautomorphism of S of order at most 2, that is (xy)* = y*x* and x** = x for all x, y ∊ S. In such a case, S is called an involutorial semigroup if regarded as a universal algebra with the binary operation of multiplication and the unary operation *. If S is also a completely simple semigroup, regarded as an algebra with multiplication and the unary operation x → x−1 of inversion (x−1 is the inverse of x in the maximal subgroup of S containing x), then (S, −1, *), or simply S, is an involutorial completely simple semigroup. All such S form a variety determined by the identities above concerning * andwhere x0 = xx−1.

scholarly journals Principal congruences in de Morgan algebras

1987 ◽  
Vol 30 (3) ◽  
pp. 415-421 ◽  
Author(s):  
M. E. Adams
Keyword(s):  
Congruence Relation ◽  
Principal Congruence ◽  
Boolean Algebras ◽  
De Morgan Algebras

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

Congruence relations, filters, ideals, and definability in lattices of α-recursively enumerable sets

1976 ◽  
Vol 41 (2) ◽  
pp. 405-418
Author(s):  
Manuel Lerman
Keyword(s):  
Congruence Relation ◽  
Lattice Theory ◽  
Elementary Theory ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Set Inclusion ◽  
Recursively Enumerable Sets ◽  
Congruence Relations ◽  
Elementary Theories

Throughout this paper, α will denote an admissible ordinal. Let (α) denote the lattice of α-r.e. sets, i.e., the lattice whose elements are the α-r.e. sets, and whose ordering is given by set inclusion. Call a set A ∈ (α)α*-finite if it is α-finite and has ordertype < α* (the Σ1-projectum of α). The α*-finite sets form an ideal of (α), and factoring (α) by this ideal, we obtain the quotient lattice *(α).We will fix a language ℒ suitable for lattice theory, and discuss decidability in terms of this language. Two approaches have succeeded in making some progress towards determining the decidability of the elementary theory of (α). Each approach was first used by Lachlan for α = ω. The first approach is to relate the decidability of the elementary theory of (α) to that of a suitable quotient lattice of (α) by a congruence relation definable in ℒ. This technique was used by Lachlan [4, §1] to obtain the equidecidability of the elementary theories of (ω) and *(ω), and was generalized by us [6, Corollary 1.2] to yield the equidecidability of the elementary theories of (α) and *(α) for all α. Lachlan [3] then adopted a different approach.

scholarly journals Rough sets based on fuzzy ideals in distributive lattices

2020 ◽  
Vol 18 (1) ◽  
pp. 122-137
Author(s):  
Yongwei Yang ◽  
Kuanyun Zhu ◽  
Xiaolong Xin
Keyword(s):  
Distributive Lattice ◽  
Rough Set ◽  
Rough Sets ◽  
Congruence Relation ◽  
Distributive Lattices ◽  
Model Based ◽  
Congruence Relations ◽  
Fuzzy Ideal ◽  
Lower And Upper Approximations ◽  
Generalized Rough Sets

Abstract In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

scholarly journals Congruence Relations for Shimura Varieties Associated withGU(n–1, 1)

2014 ◽  
Vol 66 (6) ◽  
pp. 1305-1326 ◽  
Author(s):  
Jean-Stefan Koskivirta
Keyword(s):  
Irreducible Component ◽  
Congruence Relation ◽  
Shimura Varieties ◽  
Congruence Relations ◽  
Special Case

AbstractWe prove the congruence relation for the mod–preduction of Shimura varieties associated with a unitary similitude groupGU(n– 1, 1) over ℚ whenpis inert andnodd. The case whennis even was obtained by T. Wedhorn and O. Bültel, as a special case of a result of B. Moonen, when theμ–ordinary locus of thep–isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of p–ℐsog is annihilated by a degree one polynomial in the Frobenius elementF, which implies the congruence relation.

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