Shorter Notes: The Center of a Complete Relatively Complemented Lattice is a Complete Sublattice

1967 ◽  
Vol 18 (1) ◽  
pp. 189 ◽  
Author(s):  
M. F. Janowitz
Keyword(s):  
Complete Sublattice ◽  
Complemented Lattice

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:

The Tensor Product Formula for Reflexive Subspace Lattices

1995 ◽  
Vol 38 (3) ◽  
pp. 308-316 ◽  
Author(s):  
K. J. Harrison
Keyword(s):  
Tensor Product ◽  
Operator Algebra ◽  
Product Formula ◽  
Completely Distributive ◽  
Reflexive Subspace ◽  
Complemented Lattice ◽  
Csl Algebra ◽  
Subspace Lattices

AbstractWe give a characterisation of where and are subspace lattices with commutative and either completely distributive or complemented. We use it to show that Lat is a CSL algebra with a completely distributive or complemented lattice and is any operator algebra.

The Meet Operator in the Lattice of Group Topologies

1986 ◽  
Vol 29 (4) ◽  
pp. 478-481
Author(s):  
Bradd Clark ◽  
Victor Schneider
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Locally Compact Group ◽  
Group Topology ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Lattice Of Topologies ◽  
Group Form ◽  
Complemented Lattice ◽  
Infinite Abelian Group

AbstractIt is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

scholarly journals Stabilizers in EQ-algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 998-1013
Author(s):  
Xiao Yun Cheng ◽  
Mei Wang ◽  
Wei Wang ◽  
Jun Tao Wang
Keyword(s):  
Fuzzy Logic ◽  
Left And Right ◽  
Algebraic Foundation ◽  
Complemented Lattice

Abstract The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where Str(F, G) is the relative pseudo-complemented of F with respect to G. These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.

On the notion of ranked complementedness in a lattice

2020 ◽  
pp. 2250004
Author(s):  
Francois Koch van Niekerk
Keyword(s):  
Natural Number ◽  
Modular Lattice ◽  
Finite Height ◽  
Number Formula ◽  
A Chain ◽  
Complemented Lattice

Not every element in a lattice has a complement. In this paper we introduce a notion of ranked complement, which depends on a natural number [Formula: see text], so that for every element [Formula: see text] in a lattice with finite height there exists [Formula: see text] such that [Formula: see text] has a complement of rank [Formula: see text]. One of the main results we establish is that in a modular lattice having finite height, every element has a complement of rank less than [Formula: see text] if and only if there is a chain [Formula: see text] of elements such that each interval [Formula: see text] is a complemented lattice.

scholarly journals On the lattice of congruences on an eventually regular semigroup

1985 ◽  
Vol 38 (2) ◽  
pp. 281-286 ◽  
Author(s):  
P. M. Edwards
Keyword(s):  
Regular Semigroup ◽  
Subject Classification ◽  
Maximum Element ◽  
M 10 ◽  
Natural Equivalence ◽  
20 M 10 ◽  
Complete Sublattice ◽  
Eventually Regular Semigroup ◽  
Lattice Of Congruences

AbstractA natural equivalence θ on the lattice of congruences λ(S) of a semigroup S is studied. For any eventually regular semigroup S, it is shown that θ is a congruence, each θ-class is a complete sublattice of λ(S) and the maximum element in each θ-class is determined. 1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

On the Structure of Semi-Prime Rings and their Rings of Quotients

1961 ◽  
Vol 13 ◽  
pp. 392-417 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Prime Ring ◽  
Modular Lattice ◽  
Present Author ◽  
Closure Operation ◽  
Prime Rings ◽  
Closed Submodules ◽  
Category Of Modules ◽  
Argument Proceeds ◽  
Rings Of Quotients ◽  
Complemented Lattice

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.

Multiply Subadditive Functions

1952 ◽  
Vol 4 ◽  
pp. 455-462 ◽  
Author(s):  
G. G. Lorentz
Keyword(s):  
Zero Element ◽  
Boolean Ring ◽  
Image Position ◽  
Complemented Lattice

Let S denote a Boolean ring with elements e, that is, a distributive, relatively complemented lattice with zero element 0 [2, p. 153]. In this paper we study real-valued functions which have a representation of the form1.1

The Macneille Completion of a Uniquely Complemented Lattice

1994 ◽  
Vol 37 (2) ◽  
pp. 222-227 ◽  
Author(s):  
John Harding
Keyword(s):  
Lattice Theory ◽  
Macneille Completion ◽  
The Third ◽  
Uniquely Complemented ◽  
Complemented Lattice

AbstractProblem 36 of the third edition of Birkhoff's Lattice theory [2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented.

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