scholarly journals Some operators and dimensions in modular meet-continuous lattices

2018 ◽  
Vol 17 (05) ◽  
pp. 1850094 ◽  
Author(s):  
Mauricio Medina Bárcenas ◽  
José Ríos Montes ◽  
Angel Zaldívar Corichi
Keyword(s):  
Complete Lattice ◽  
Monotone Function ◽  
Continuous Lattice ◽  
Continuous Lattices

Given a complete modular meet-continuous lattice [Formula: see text], an inflator on [Formula: see text] is a monotone function [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is the set of all inflators on [Formula: see text], then [Formula: see text] is a complete lattice. Motivated by preradical theory, we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice [Formula: see text] and with the concept of dimension.

A completion-invariant extension of the concept of meet continuous lattices

2015 ◽  
Vol 27 (4) ◽  
pp. 530-539
Author(s):  
WENFENG ZHANG ◽  
XIAOQUAN XU
Keyword(s):  
Heyting Algebra ◽  
Continuous Lattice ◽  
Stable Maps ◽  
Complete Lattices ◽  
Invariant Extension ◽  
Closed Sets ◽  
Reflective Subcategory ◽  
Normal Completion ◽  
Continuous Lattices ◽  
Algebra 2

In this paper, the concept of meet F-continuous posets is introduced. The main results are: (1) A poset P is meet F-continuous iff its normal completion is a meet continuous lattice iff a certain system γ(P) which is, in the case of complete lattices, the lattice of all Scott closed sets is a complete Heyting algebra; (2) A poset P is precontinuous iff P is meet F-continuous and quasiprecontinuous; (3) The category of meet continuous lattices with complete homomorphisms is a full reflective subcategory of the category of meet F-continuous posets with cut-stable maps.

scholarly journals A completion-invariant extension of the concept of quasi C-continuous lattices

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2345-2353 ◽  
Author(s):  
Xiaojun Ruan ◽  
Xiaoquan Xu
Keyword(s):  
Continuous Lattice ◽  
Completely Continuous ◽  
Invariant Extension ◽  
Normal Completion ◽  
Continuous Lattices

In this paper, the concepts of C-precontinuous posets, quasi C-precontinuous posets and meet Cprecontinuous posets are introduced. The main results are: (1) A complete semilattice P is C-precontinuous (resp., quasi C-precontinuous) if and only if its normal completion is a C-continuous lattice (resp., quasi C-continuous lattice); (2) A poset is both quasi C-precontinuous and Frink quasicontinuous if and only if it is generalized completely continuous; (3) A complete semilattice is meet C-precontinuous if and only if its normal completion is meet C-continuous; (4) A poset is both quasi C-precontinuous and meet C-precontinuous if and only if it is C-precontinuous.

The Duality of Distributive Continuous Lattices

1980 ◽  
Vol 32 (2) ◽  
pp. 385-394 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Dual Space ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Continuous Lattice ◽  
Prime Spectrum ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Lattices ◽  
Prime Elements

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

scholarly journals Fixpoint Theory – Upside Down

2021 ◽  
pp. 62-81
Author(s):  
Paolo Baldan ◽  
Richard Eggert ◽  
Barbara König ◽  
Tommaso Padoan
Keyword(s):  
Complete Lattice ◽  
Stochastic Games ◽  
Monotone Function ◽  
Monotone Functions ◽  
Probabilistic Automata ◽  
Proof Rules ◽  
Wide Range ◽  
Mv Algebra ◽  
Finite Set ◽  
Inductive Invariants

AbstractKnaster-Tarski’s theorem, characterising the greatest fix- point of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $$\mathbb {M}^Y$$ M Y , where Y is a finite set and $$\mathbb {M}$$ M an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

scholarly journals Modeling Imprecise and Bipolar Algebraic and Topological Relations using Morphological Dilations

2021 ◽  
Vol 5 (1) ◽  
pp. 1-20
Author(s):  
Isabelle Bloch
Keyword(s):  
Information Processing ◽  
Fuzzy Sets ◽  
Mathematical Morphology ◽  
Complete Lattice ◽  
Spatial Reasoning ◽  
Topological Relations ◽  
Preference Modeling ◽  
Logical Sense ◽  
Bipolar Fuzzy Sets ◽  
Core Features

Abstract In many domains of information processing, such as knowledge representation, preference modeling, argumentation, multi-criteria decision analysis, spatial reasoning, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools, such as fusion and aggregation, similarity and distances, mathematical morphology. Here we propose to extend these tools by defining algebraic and topological relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators. These definitions are shown to have the desired properties and to be consistent with existing definitions on sets and fuzzy sets, while providing an additional bipolar feature. The proposed relations can be used for instance for preference modeling or spatial reasoning. They apply more generally to any type of functions taking values in a poset or a complete lattice, such as L-fuzzy sets.

Induced L-bornological vector spaces and L-Mackey convergence1

2021 ◽  
Vol 40 (1) ◽  
pp. 1277-1285
Author(s):  
Zhen-yu Jin ◽  
Cong-hua Yan
Keyword(s):  
Complete Lattice ◽  
Vector Spaces ◽  
Specific Description ◽  
Equivalent Characterization ◽  
Fuzzy Setting

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

Fuzzy deductive systems of RM algebras

2020 ◽  
Vol 70 (6) ◽  
pp. 1259-1274
Keyword(s):  
Fuzzy Set ◽  
Complete Lattice ◽  
Deductive System ◽  
Deductive Systems

AbstractThe theory of fuzzy deductive systems in RM algebras is developed. Various characterizations of fuzzy deductive systems are given. It is proved that the set of all fuzzy deductive systems of a RM algebra 𝒜 is a complete lattice (it is distributive if 𝒜 is a pre-BBBCC algebra). Some characterizations of Noetherian RM algebras by fuzzy deductive systems are obtained. In pre-BBBZ algebras, the fuzzy deductive system generated by a fuzzy set is constructed. Finally, closed fuzzy deductive systems are defined and studied. It is showed that in finite CI and pre-BBBZ algebras, every fuzzy deductive system is closed. Moreover, the homomorphic properties of (closed) fuzzy deductive systems are provided.

Linear types and approximation

2000 ◽  
Vol 10 (6) ◽  
pp. 719-745 ◽  
Author(s):  
MICHAEL HUTH ◽  
ACHIM JUNG ◽  
KLAUS KEIMEL
Keyword(s):  
Full Subcategory ◽  
Linear Logic ◽  
Domain Theory ◽  
Classical Domain ◽  
Complete Distributivity ◽  
Linear Types ◽  
Continuous Lattices

We study continuous lattices with maps that preserve all suprema rather than only directed ones. We introduce the (full) subcategory of FS-lattices, which turns out to be *-autonomous, and in fact maximal with this property. FS-lattices are studied in the presence of distributivity and algebraicity. The theory is extremely rich with numerous connections to classical Domain Theory, complete distributivity, Topology and models of Linear Logic.

Sign in / Sign up

Export Citation Format

Share Document