Displaying similar documents to “Using lattice homomorphisms in decomposition theory”

Lattice of ℤ-module

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding...

Dual Lattice of ℤ-module Lattice

Yuichi Futa, Yasunari Shidama (2017)

Formalized Mathematics

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In this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic...

Classification systems and their lattice

Sándor Radeleczki (2002)

Discussiones Mathematicae - General Algebra and Applications

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We define and study classification systems in an arbitrary CJ-generated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(L)). If B(L) is a pseudocomplemented lattice, then every classification system of L is independent...

On skew lattices. II.

Václav Slavík (1973)

Commentationes Mathematicae Universitatis Carolinae

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Embedded Lattice and Properties of Gram Matrix

Yuichi Futa, Yasunari Shidama (2017)

Formalized Mathematics

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In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].

Residuation in orthomodular lattices

Ivan Chajda, Helmut Länger (2017)

Topological Algebra and its Applications

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We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one. ...

Finite atomistic lattices that can be represented as lattices of quasivarieties

K. Adaricheva, Wiesław Dziobiak, V. Gorbunov (1993)

Fundamenta Mathematicae

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We prove that a finite atomistic lattice can be represented as a lattice of quasivarieties if and only if it is isomorphic to the lattice of all subsemilattices of a finite semilattice. This settles a conjecture that appeared in the context of [11].