scholarly journals Divisible ℤ-modules

2016 ◽  
Vol 24 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Coefficient Ring ◽  
Torsion Free

Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

scholarly journals Lattice of ℤ-module

2016 ◽  
Vol 24 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Coding Theory ◽  
Positive Definite ◽  
Bilinear Forms ◽  
Reduction Algorithm ◽  
Real Numbers ◽  
Scalar Products ◽  
Definition Of

Summary 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 theory [9].

scholarly journals Torsion Z-module and Torsion-free Z-module

2014 ◽  
Vol 22 (4) ◽  
pp. 277-289
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Finitely Generated ◽  
Torsion Free

Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

scholarly journals Torsion Part of ℤ-module

2015 ◽  
Vol 23 (4) ◽  
pp. 297-307 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Rational Numbers ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Torsion Part ◽  
Torsion Free

Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

New definition of MV-semimodules

2021 ◽  
pp. 1-12
Author(s):  
Simin Saidi Goraghani ◽  
Rajab Ali Borzooei ◽  
Sun Shin Ahn
Keyword(s):  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Torsion Free

In recent years, A. Di Nola et al. studied the notions of MV-semiring and semimodules and investigated related results [9, 10, 12, 26]. Now in this paper, by using an MV-semiring and an MV-algebra, we introduce the new definition of MV-semimodule, study basic properties and find some examples. Then we study A-ideals on MV-semimodules and Q-ideals on MV-semirings, and by using them, we study the quotient structures of MV-semimodule. Finally, we present the notions of prime A-ideal, torsion free MV-semimodule and annihilator on MV-semimodule and we study the relations among them.

ON FINITE TYPE 3-MANIFOLD INVARIANTS I

1996 ◽  
Vol 05 (04) ◽  
pp. 441-461 ◽  
Author(s):  
STAVROS GAROUFALIDIS
Keyword(s):  
Vector Space ◽  
Finite Type ◽  
Commutative Algebra ◽  
Integral Homology ◽  
Knot Invariants ◽  
Finite Type Invariants ◽  
Definition Of ◽  
Type 3

Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.

scholarly journals Submodule of free Z-module

2013 ◽  
Vol 21 (4) ◽  
pp. 273-282 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Reduction Algorithm ◽  
Rational Field

Abstract In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.

Optimization Analysis on Dynamic Reduction Algorithm

2018 ◽  
Vol 6 (5) ◽  
pp. 447-458
Author(s):  
Yizhou Chen ◽  
Jiayang Wang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Reduction Process ◽  
New Method ◽  
Reduction Algorithm ◽  
Optimization Analysis ◽  
Sampling Process ◽  
Dynamic Methods ◽  
Definition Of

Abstract On the basis of rough set theory, the strengths of dynamic reduction are elaborated compared with traditional non-dynamic methods. A systematic concept of dynamic reduction from sampling process to the generation of the reduct set is presented. A new method of sampling is created to avoid the defects of being too subjective. And in order to deal with the over-sized time consuming problem in traditional dynamic reduction process, a quick algorithm is proposed within the constraint conditions. We have also proved that dynamic core possesses the essential characteristics of a reduction core on the basis of the formalized definition of the multi-layered dynamic core.

Semiring of Generalized Interval-Valued Intuitionistic Fuzzy Matrices

2017 ◽  
pp. 132-152
Author(s):  
Debashree Manna
Keyword(s):  
Vector Space ◽  
Distributive Lattice ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Algebra ◽  
Definition Of ◽  
Interval Valued

In this paper, the concept of semiring of generalized interval-valued intuitionistic fuzzy matrices are introduced and have shown that the set of GIVIFMs forms a distributive lattice. Also, prove that the GIVIFMs form an generalized interval valued intuitionistic fuzzy algebra and vector space over [0, 1]. Some properties of GIVIFMs are studied using the definition of comparability of GIVIFMs.

scholarly journals On Rotor calculus, I

1966 ◽  
Vol 6 (4) ◽  
pp. 402-423 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Vector Space ◽  
Lorentz Transformation ◽  
Real World ◽  
Three Dimensional ◽  
Linear Vector ◽  
Complex Rotation ◽  
Starting Point ◽  
Complex Linear ◽  
Definition Of ◽  
Linear Vector Space

SummaryIt is known that to every proper homogeneous Lorentz transformation there corresponds a unique proper complex rotation in a three-dimensional complex linear vector space, the elements of which are here called “rotors”. Equivalently one has a one-one correspondence between rotors and self- dual bi-vectors in space-time (w-space). Rotor calculus fully exploits this correspondence, just as spinor calculus exploits the correspondence between real world vectors and hermitian spinors; and its formal starting point is the definition of certain covariant connecting quantities τAkl which transform as vectors under transformations in rotor space (r-space) and as tensors of valence 2 under transformations in w-space.

Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

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