The least extension of a closure operator on a sublattice

1967 ◽  
Vol 63 (1) ◽  
pp. 9-10 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Complete Lattice ◽  
Closure Operator ◽  
Partial Ordering ◽  
Closure Operators ◽  
Image Position

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditionsfor all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

scholarly journals Regular ω-semigroups

1968 ◽  
Vol 9 (1) ◽  
pp. 46-66 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Partial Ordering ◽  
Image Position

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chainthen S is called an ω-semigroup.

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

scholarly journals The lateral completion of an arbitrary lattice group

1975 ◽  
Vol 19 (3) ◽  
pp. 263-289 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Complete Lattice ◽  
Lattice Group ◽  
Arbitrary Lattice ◽  
Image Position ◽  
Structure Properties ◽  
The Ideal

SummaryThis paper shows that every lattice group G can be densely embedded in a unique laterally complete lattice group H (the lateral completion of G). All reasonable structure properties of G are inherited by H and we have the following relationships between the ideal radical L(G), the distributive radical D(G) and the radical R(G) of G and the corresponding radicals of H. .

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

A lattice of interpretability types of theories

1977 ◽  
Vol 42 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Order Theory ◽  
Partial Ordering ◽  
First Order ◽  
First Order Theory ◽  
Relative Consistency ◽  
Infinite Distributivity ◽  
Closed Fields ◽  
Image Position ◽  
Interpretability Type ◽  
Consistency Proofs

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

scholarly journals Path-set induced closure operators on graphs

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 863-871 ◽  
Author(s):  
Josef Slapal
Keyword(s):  
Image Analysis ◽  
Digital Image ◽  
Digital Image Analysis ◽  
Closure Operator ◽  
Simple Graph ◽  
Digital Topology ◽  
Closure Operators ◽  
Vertex Set ◽  
Positive Length ◽  
Path Connectedness

Given a simple graph, we associate with every set of paths of the same positive length a closure operator on the (vertex set of the) graph. These closure operators are then studied. In particular, it is shown that the connectedness with respect to them is a certain kind of path connectedness. Closure operators associated with sets of paths in some graphs with the vertex set Z2 are discussed which include the well known Marcus-Wyse and Khalimsky topologies used in digital topology. This demonstrates possible applications of the closure operators investigated in digital image analysis.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

Generalized Fuzzy Closed Sets in Smooth Bitopological Spaces

2016 ◽  
pp. 419-456
Author(s):  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh ◽  
Rasha N. Majeed
Keyword(s):  
Closure Operator ◽  
Closure Operators ◽  
Closed Sets ◽  
Different Types ◽  
Definition Of ◽  
Bitopological Spaces

This chapter is devoted to the study of r-generalized fuzzy closed sets (briefly, gfc sets) in smooth bitopological spaces (briefly, smooth bts) in view definition of Šostak (1985). The chapter is divided into seven sections. The aim of Sections 1-2 is to introduce the fundamental concepts related to the work. In Section 3, the concept of r-(ti,tj)-gfc sets in the smooth bts's is introduce and investigate some notions of these sets, generalized fuzzy closure operator induced from these sets. In Section 4, (i,j)-GF-continuous (respectively, irresolute) mappings are introduced. In Section 5, the supra smooth topology which generated from a smooth bts is used to introduce and study the notion of r-t12-gfc sets for smooth bts's and supra generalized fuzzy closure operator . The present notion of gfc sets and the notion which introduced in section 3, are independent. In Section 6, two types of generalized supra fuzzy closure operators introduced by using two different approaches. Finally, Section 7 introduces and studies different types of fuzzy continuity which are related to closure.

Sign in / Sign up

Export Citation Format

Share Document