$G$-supplemented property in the lattices

Shahabaddin Ebrahimi Atani

Displaying similar documents to “ $G$ -supplemented property in the lattices”

Relative co-annihilators in lattice equality algebras

Sogol Niazian, Mona Aaly Kologani, Rajab Ali Borzooei (2024)

Mathematica Bohemica

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We introduce the notion of relative co-annihilator in lattice equality algebras and investigate some important properties of it. Then, we obtain some interesting relations among $\lor$ -irreducible filters, positive implicative filters, prime filters and relative co-annihilators. Given a lattice equality algebra $𝔼$ and $𝔽$ a filter of $𝔼$ , we define the set of all $𝔽$ -involutive filters of $𝔼$ and show that by defining some operations on it, it makes a BL-algebra.

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.

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...

A new approach to construct uninorms via uninorms on bounded lattices

Zhen-Yu Xiu, Xu Zheng (2024)

Kybernetika

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In this paper, on a bounded lattice $L$ , we give a new approach to construct uninorms via a given uninorm $U^{*}$ on the subinterval $[0, a]$ (or $[b, 1]$ ) of $L$ under additional constraint conditions on $L$ and $U^{*}$ . This approach makes our methods generalize some known construction methods for uninorms in the literature. Meanwhile, some illustrative examples for the construction of uninorms on bounded lattices are provided.

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...

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

Lifeng Li, Jianke Zhang, Chang Zhou (2019)

Kybernetika

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For a t-norm T on a bounded lattice $(L, \leq)$ , a partial order $\leq_{T}$ was recently defined and studied. In [11], it was pointed out that the binary relation $\leq_{T}$ is a partial order on $L$ , but $(L, \leq_{T})$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L, \leq_{T})$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L, \leq_{T})$ is a lattice are presented.

Modular lattices from finite projective planes

Tathagata Basak (2014)

Journal de Théorie des Nombres de Bordeaux

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Using the geometry of the projective plane over the finite field $𝔽_{q}$ , we construct a Hermitian Lorentzian lattice $L_{q}$ of dimension $(q^{2} + q + 2)$ defined over a certain number ring $𝒪$ that depends on $q$ . We show that infinitely many of these lattices are $p$ -modular, that is, $p L_{q}^{'} = L_{q}$ , where $p$ is some prime in $𝒪$ such that ${| p |}^{2} = q$ . The Lorentzian lattices $L_{q}$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q \equiv 3 mod 4$ is a rational prime such that $(q^{2} + q + 1)$ is norm of some element in...

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 ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

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....

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

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In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if $σ \subseteq D \in M$ is a support, $M$ being an inner model of ZFC, and $𝒫 (D ∖ σ) \cap M = r ` ` σ$ with $r \in M$ , then $r$ determines a preorder " $⪯$ " of $D$ such that $σ$ becomes a filter on $(D, ⪯)$ generic over $M$ . We show that if the relation $r$ is replaced by a function $𝒫 (D ∖ σ) \cap M = f_{- 1} (σ)$ , then there exists an equivalence relation " $\sim$ " on $D$ and a partial order on $D / \sim$ such that $D / \sim$ is a complete Boolean algebra, $σ / \sim$ is a generic filter and ${[f (u)]}_{\sim} = - \sum (u / \sim)$ for...

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 $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

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...

Some properties of state filters in state residuated lattices

Michiro Kondo (2021)

Mathematica Bohemica

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We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$ : (1) $F$ ...

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. ...

Displaying similar documents to “ G -supplemented property in the lattices”

Displaying similar documents to “ $G$ -supplemented property in the lattices”