Formalized Mathematics最新文献_第2页

Elementary Number Theory Problems. Part VII 初等数论问题。第七部分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0003

Artur Korniłowicz

引用次数: 0

Elementary Number Theory Problems. Part XI 初等数论问题。第十一部分

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0021

Adam Naumowicz

引用次数: 0

Internal Direct Products and the Universal Property of Direct Product Groups 内部直接积与直接积群的全称性质

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0010

Alexander M. Nelson

{"title":"Internal Direct Products and the Universal Property of Direct Product Groups","authors":"Alexander M. Nelson","doi":"10.2478/forma-2023-0010","DOIUrl":"https://doi.org/10.2478/forma-2023-0010","url":null,"abstract":"Abstract This is a “quality of life” article concerning product groups, using the Mizar system [2], [4]. Like a Sonata, this article consists of three movements. The first act, the slowest of the three, builds the infrastructure necessary for the rest of the article. We prove group homomorphisms map arbitrary finite products to arbitrary finite products, introduce a notion of “group yielding” families, as well as families of homomorphisms. We close the first act with defining the inclusion morphism of a subgroup into its parent group, and the projection morphism of a product group onto one of its factors. The second act introduces the universal property of products and its consequences as found in, e.g., Kurosh [7]. Specifically, for the product of an arbitrary family of groups, we prove the center of a product group is the product of centers. More exciting, we prove for a product of a finite family groups, the commutator subgroup of the product is the product of commutator subgroups, but this is because in general: the direct sum of commutator subgroups is the subgroup of the commutator subgroup of the product group, and the commutator subgroup of the product is a subgroup of the product of derived subgroups. We conclude this act by proving a few theorems concerning the image and kernel of morphisms between product groups, as found in Hungerford [5], as well as quotients of product groups. The third act introduces the notion of an internal direct product. Isaacs [6] points out (paraphrasing with Mizar terminology) that the internal direct product is a predicate but the external direct product is a [Mizar] functor. To our delight, we find the bulk of the “recognition theorem” (as stated by Dummit and Foote [3], Aschbacher [1], and Robinson [11]) are already formalized in the heroic work of Nakasho, Okazaki, Yamazaki, and Shimada [9], [8]. We generalize the notion of an internal product to a set of subgroups, proving it is equivalent to the internal product of a family of subgroups [10].","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135736687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Differentiation on Interval 区间微分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0002

Noboru Endou

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引用次数: 1

Introduction to Graph Enumerations 图枚举介绍

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0004

Sebastian Koch

引用次数: 0

Integration of Game Theoretic and Tree Theoretic Approaches to Conway Numbers 将博弈论和树论方法整合到康威数中

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0019

Karol Pąk

引用次数: 0

Elementary Number Theory Problems. Part X – Diophantine Equations 初等数论问题。第十部分 - Diophantine方程

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0016

Artur Korniłowicz

引用次数: 0

On Bag of 1. Part I 一袋1。第一部分

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0001

Yasushige Watase

{"title":"On Bag of 1. Part I","authors":"Yasushige Watase","doi":"10.2478/forma-2023-0001","DOIUrl":"https://doi.org/10.2478/forma-2023-0001","url":null,"abstract":"Summary The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable in terms of “bag” (as described in detail in [9]), the same notion as multiset over a finite set, in the Mizar system [1], [2]. Polynomial rings and ring of formal power series, both in one variable, have been formalized in [6], [5] respectively, and elements of these rings are represented by infinite sequences of scalars. On the other hand, formalization of a multivariate polynomial requires extra techniques of using “bag” to represent monomials of variables, and polynomials are formalized as a function from bags of variables to the scalar ring. This means the way of construction of the rings are different between single variable and multi variables case (which implies some tedious constructions, e.g. in the case of ten variables in [8], or generally in the problem of prime representing polynomial [7]). Introducing bag-based construction to one variable polynomial ring provides straight way to apply mathematical induction to polynomial rings with respect to the number of variables. Another consequence from the article, a polynomial ring is a subring of an algebra [4] over the same scalar ring, namely a corresponding formal power series. A sketch of actual formalization of the article is consists of the following four steps: 1. translation between Bags 1 (the set of all bags of a singleton) and N; 2. formalization of a bag-based formal power series in multivariable case over a commutative ring denoted by Formal-Series ( n, R ); 3. formalization of a polynomial ring in one variable by restricting one variable case denoted by Polynom-Ring (1, R ). A formal proof of the fact that polynomial rings are a subring of Formal-Series ( n, R ), that is R -Algebra, is included as well; 4. formalization of a ring isomorphism to the existing polynomial ring in one variable given by sequence: Polynom-Ring (1, R ) →˜ Polynom-Ring .","PeriodicalId":42667,"journal":{"name":"Formalized Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134995200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Elementary Number Theory Problems. Part IX 初等数论问题。第九部分

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0015

Artur Korniłowicz

引用次数: 0

The Ring of Conway Numbers in Mizar 米扎尔的康威数字之环

IF 0.3

Formalized Mathematics Pub Date : 2023-09-01 DOI: 10.2478/forma-2023-0020

Karol Pąk

引用次数: 0