Goldie extending elements in modular lattices

Shriram K. Nimbhorkar; Rupal C. Shroff

Displaying similar documents to “Goldie extending elements in modular lattices”

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.

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.

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...

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...

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.

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.

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

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...

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman (2020)

Czechoslovak Mathematical Journal

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Let $H$ denote a finite index subgroup of the modular group $Γ$ and let $ρ$ denote a finite-dimensional complex representation of $H .$ Let $M (ρ)$ denote the collection of holomorphic vector-valued modular forms for $ρ$ and let $M (H)$ denote the collection of modular forms on $H$ . Then $M (ρ)$ is a $ℤ$ -graded $M (H)$ -module. It has been proven that $M (ρ)$ may not be projective as a $M (H)$ -module. We prove that $M (ρ)$ is Cohen-Macaulay as a $M (H)$ -module. We also explain how to apply this result to prove that if $M (H)$ is a polynomial ring, then...

Wallman-type compaerifications and function lattices

Alessandro Caterino, Maria Cristina Vipera (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Let $F\subset C^{\ast}(X)$ be a vector sublattice over $\mathbb{R}$ which separates points from closed sets of $X$ . The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega(Z(F))$ . In this paper, it is shown that there exists a lattice $F_{z}$ containing $F$ such that $\omega(Z(F))=\omega(Z(F_{z}))=e_{F}X$ . In particular this implies that $\omega(Z(F))\geq e_{F}X$ . Conditions in order to be $\omega(Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $Cl_{\alpha X}(\alpha X\setminus X)$ is $0$ -dimensional, then there is an...

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 ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$ ; the $c$ -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$ . Its maximal cones are naturally indexed by the $c$ -sortable elements of $W$ . The main result of this paper is that the known bijection cl $_{c}$ between $c$ -sortable elements and $c$ -clusters induces a combinatorial isomorphism of fans. In particular, the $c$ -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...