Residuated Operators and Dedekind–MacNeille Completion

G. Czédli’s tolerance factor lattice construction and weak ordered relations

Algebra Universalis ◽

10.1007/s00012-021-00712-x ◽

2021 ◽

Vol 82 (2) ◽

Author(s):

Sándor Radeleczki

Keyword(s):

Concept Lattice ◽

Tolerance Factor ◽

Macneille Completion ◽

Lattice Construction

AbstractG. Czédli proved that the blocks of any compatible tolerance T of a lattice L can be ordered in such a way that they form a lattice L/T called the factor lattice of L modulo T. Here we show that the Dedekind–MacNeille completion of the lattice L/T is isomorphic to the concept lattice of the context (L, L, R), where R stands for the reflexive weak ordered relation $$ \mathord {\le } \circ T$$ ≤ ∘ T . Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations $$R\subseteq L^{2}$$ R ⊆ L 2 satisfying $$R=\ \mathord {\le } \circ R\circ \mathord {\le } $$ R = ≤ ∘ R ∘ ≤ .

A generalization of the Dedekind–MacNeille completion

Semigroup Forum ◽

10.1007/s00233-017-9911-4 ◽

2017 ◽

Vol 96 (3) ◽

pp. 553-564 ◽

Cited By ~ 3

Author(s):

Zhongxi Zhang ◽

Qingguo Li

Keyword(s):

Macneille Completion

Generalized dimension of an ordered set and its MacNeille completion

Order ◽

10.1007/bf01108598 ◽

1994 ◽

Vol 11 (2) ◽

pp. 135-148

Author(s):

Philippe Baldy ◽

Jutta Mitas

Keyword(s):

Ordered Set ◽

Macneille Completion ◽

Generalized Dimension

Betweenness relations and cycle-free partial orders

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100074478 ◽

1996 ◽

Vol 119 (4) ◽

pp. 631-643 ◽

Cited By ~ 7

Author(s):

J. K. Truss

Keyword(s):

Partial Order ◽

Linear Order ◽

Partial Orders ◽

Partial Ordering ◽

Macneille Completion ◽

Alternative Definition ◽

Betweenness Relations

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

On the Dedekind–MacNeille completion and formal concept analysis based on multilattices

Fuzzy Sets and Systems ◽

10.1016/j.fss.2016.01.007 ◽

2016 ◽

Vol 303 ◽

pp. 1-20 ◽

Cited By ~ 6

Author(s):

J. Medina ◽

M. Ojeda-Aciego ◽

J. Pócs ◽

E. Ramírez-Poussa

Keyword(s):

Formal Concept Analysis ◽

Concept Analysis ◽

Formal Concept ◽

Macneille Completion

On the MacNeille Completion of the Category of Partially Ordered Topological Spaces

Mathematische Nachrichten ◽

10.1002/mana.19931630123 ◽

1993 ◽

Vol 163 (1) ◽

pp. 281-288 ◽

Cited By ~ 2

Author(s):

A. Schauerte

Keyword(s):

Topological Spaces ◽

Macneille Completion ◽

Partially Ordered

A Syntactic Approach to the MacNeille Completion of Λ∗, the Free Monoid Over an Ordered Alphabet Λ

Order ◽

10.1007/s11083-018-9462-7 ◽

2018 ◽

Vol 36 (2) ◽

pp. 209-224 ◽

Cited By ~ 1

Author(s):

Hans-Jürgen Bandelt ◽

Maurice Pouzet

Keyword(s):

Free Monoid ◽

Macneille Completion ◽

Syntactic Approach

The Macneille Completion of a Uniquely Complemented Lattice

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-033-3 ◽

1994 ◽

Vol 37 (2) ◽

pp. 222-227 ◽

Cited By ~ 2

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.

Dedekind-MacNeille completion of multivariate copulas via ALGEN method

Fuzzy Sets and Systems ◽

10.1016/j.fss.2021.10.011 ◽

2021 ◽

Author(s):

Matjaž Omladič ◽

Nik Stopar

Keyword(s):

Macneille Completion

Endpoints inT0-Quasimetric Spaces: Part II

Abstract and Applied Analysis ◽

10.1155/2013/539573 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 5

Author(s):

Collins Amburo Agyingi ◽

Paulus Haihambo ◽

Hans-Peter A. Künzi

Keyword(s):

Complete Lattice ◽

Partially Ordered Set ◽

Ordered Set ◽

Partial Orders ◽

Macneille Completion ◽

Partially Ordered ◽

Irreducible Elements ◽

Quasimetric Space

We continue our work on endpoints and startpoints inT0-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valuedT0-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and theq-hyperconvex hull of its naturalT0-quasimetric space.