Some Remarks About Idempotent Uninorms on Complete Lattice

2017 ◽  
pp. 648-657 ◽  
Author(s):  
Paweł Drygaś
Keyword(s):  
Complete Lattice

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.

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.

scholarly journals On the Structure of Conditionally Complete Lattice-Groups

1941 ◽  
Vol 18 (0) ◽  
pp. 777-789 ◽  
Author(s):  
Kenkichi IWASAWA
Keyword(s):  
Complete Lattice

Existentially complete lattice-ordered groups

1980 ◽  
Vol 36 (3-4) ◽  
pp. 257-272 ◽  
Author(s):  
A. M. W. Glass ◽  
Keith R. Pierce
Keyword(s):  
Complete Lattice ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Sylow Theory for a Certain Class of Operator Groups

1961 ◽  
Vol 13 ◽  
pp. 192-200 ◽  
Author(s):  
Christine W. Ayoub
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Complete Lattice ◽  
Classical Group ◽  
Additive Group ◽  
Sylow Subgroups ◽  
Direct Sum

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

scholarly journals Divisibility of ordered groups

1972 ◽  
Vol 18 (1) ◽  
pp. 81-83
Author(s):  
I. W. Wright
Keyword(s):  
Abelian Group ◽  
Complete Lattice ◽  
Positive Cone ◽  
Real Numbers ◽  
Ordered Groups ◽  
Small Elements

In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone.We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as defined in Birkhoff(1). For the reader's convenience we now define the two terms most used in the sequel.

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