Sufficient conditions for a T-partial order obtained from triangular norms to be a lattice

Lifeng Li; Jianke Zhang; Chang Zhou

Displaying similar documents to “Sufficient conditions for a T-partial order obtained from triangular norms to be a lattice”

Orthogonality and complementation in the lattice of subspaces of a finite vector space

Ivan Chajda, Helmut Länger (2022)

Mathematica Bohemica

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We investigate the lattice $𝐋 (𝐕)$ of subspaces of an $m$ -dimensional vector space $𝐕$ over a finite field $GF (q)$ with a prime power $q = p^{n}$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋 (𝐕)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋 (𝐕)$ is orthomodular....

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

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Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_{3}$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset $J (Con L)$ of join-irreducible congruences of $L$ . We prove that...

Hyperreflexivity of bilattices

Kamila Kliś-Garlicka (2016)

Czechoslovak Mathematical Journal

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The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $ℒ$ we can construct the bilattice $Σ_{ℒ}$ . Similarly, having a bilattice $Σ$ we may consider the lattice $ℒ_{Σ}$ . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples...

Construction of uninorms on bounded lattices

Gül Deniz Çaylı, Funda Karaçal (2017)

Kybernetika

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In this paper, we propose the general methods, yielding uninorms on the bounded lattice $(L, \leq, 0, 1)$ , with some additional constraints on $e \in L ∖ {0, 1}$ for a fixed neutral element $e \in L ∖ {0, 1}$ based on underlying an arbitrary triangular norm $T_{e}$ on $[0, e]$ and an arbitrary triangular conorm $S_{e}$ on $[e, 1]$ . And, some illustrative examples are added for clarity.

Some methods to obtain t-norms and t-conorms on bounded lattices

Gül Deniz Çaylı (2019)

Kybernetika

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In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice $L$ based on a priori given t-norm acting on $[a, 1]$ and t-conorm acting on $[0, a]$ for an arbitrary element $a \in L ∖ {0, 1}$ . We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice. ...

Goldie extending elements in modular lattices

Shriram K. Nimbhorkar, Rupal C. Shroff (2017)

Mathematica Bohemica

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The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b \leq a$ there exists a direct summand $c$ of $a$ such that $b \land c$ is essential in both $b$ and $c$ . Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the $ℤ^{d}$ case and obtain an explicit construction of a generic point of the $ℤ^{d}$ shift transformation of the set ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

Relations on a lattice of varieties of completely regular semigroups

Mario Petrich (2020)

Mathematica Bohemica

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Completely regular semigroups $𝒞ℛ$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $𝒞ℛ$ a variety; its lattice of subvarieties is denoted by $ℒ (𝒞ℛ)$ . We study here the relations $𝐊, T, L$ and $𝐂$ relative to a sublattice $Ψ$ of $ℒ (𝒞ℛ)$ constructed in a previous publication. For $𝐑$ being any of these relations, we determine the $𝐑$ -classes of all varieties in the lattice $Ψ$ as well as the restrictions of $𝐑$ to $Ψ$ .

Quasicontinuous spaces

Jing Lu, Bin Zhao, Kaiyun Wang, Dong Sheng Zhao (2022)

Commentationes Mathematicae Universitatis Carolinae

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We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_{0}$ space $(X, τ)$ is a quasicontinuous space if and only if $S I (X)$ is locally hypercompact if and only if $(τ_{S I}, \subseteq)$ is a hypercontinuous lattice; (2) a $T_{0}$ space $X$ is an $S I$ -continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$ -space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous...

Lattice copies of c₀ and $ℓ^{\infty}$ in spaces of integrable functions for a vector measure

S. Okada, W. J. Ricker, E. A. Sánchez Pérez

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The spaces L¹(m) of all m-integrable (resp. $L ¹_{w} (m)$ of all scalarly m-integrable) functions for a vector measure m, taking values in a complex locally convex Hausdorff space X (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, $L ¹_{w} (m)$ is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(m) is also a complex vector lattice is one of our central aims. Whenever X is sequentially complete, then this is the case. If,...

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

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Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , where the sum is taken over the lattice of all points k in $ℝ^{n}$ having integer-valued components, j∈ℕ and $a_{k} \in ℂ$ . Let $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

Order intervals in $C (K)$ . Compactness, coincidence of topologies, metrizability

Zbigniew Lipecki (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be a compact space and let $C (K)$ be the Banach lattice of real-valued continuous functions on $K$ . We establish eleven conditions equivalent to the strong compactness of the order interval $[0, x]$ in $C (K)$ , including the following ones: (i) ${x > 0}$ consists of isolated points of $K$ ; (ii) $[0, x]$ is pointwise compact; (iii) $[0, x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0, x]$ ; (v) the strong and weak topologies coincide on $[0, x]$ . Moreover, the weak topology and that of pointwise...

$G$ -supplemented property in the lattices

Shahabaddin Ebrahimi Atani (2022)

Mathematica Bohemica

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Let $L$ be a lattice with the greatest element $1$ . Following the concept of generalized small subfilter, we define $g$ -supplemented filters and investigate the basic properties and possible structures of these filters.

Direct summands of Goldie extending elements in modular lattices

Rupal Shroff (2022)

Mathematica Bohemica

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In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b \leq a$ there exists a direct summand $c$ of $a$ such that $b \land c$ is essential in both $b$ and $c$ . Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.