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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Vector spaces without fields ： ウィキペディア英語版 Vector spaces without fields In mathematics, the conventional definition of the concept of vector space relies upon the algebraic concept of a field. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as (universal) algebras as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such vector space algebras. One of the ways to do this is to keep the first four abelian group axioms on addition in the standard formal definition and to formalize its ''geometric'' idea of scaling only by notions that concern every universal algebra.〔The “generalized conception of space” in the preface of is the first published claim that geometric ideas concern Universal Algebra too. It expands a similar claim in the preface of .〕 Vector space algebras consist of ''one'' binary operation "+" and of ''unary'' operations , which form a ''nonempty'' set , that satisfy the following conditions, which do not involve fields. # (''Total homogeneous algebra'') There is a single set such that every operation takes its two arguments or its argument from the whole and gives its value in it. # (''Abelian group'') + satisfies the above mentioned axioms. # (''Basis dilation'') There is a basis set $B\backslash subseteq\; V$ such that, for every such that is not constant, all the values $\backslash delta(b)$, where ranges over , again form a basis set. # (''Dilatations'') is the set of functions that satisfy the previous conditions and ''preserve'' all operations, namely $\backslash delta(v+w)=\backslash delta(v)+\backslash delta(w)$ and $\backslash delta(\backslash gamma(v))=\backslash gamma(\backslash delta(v))$, for all $\backslash gamma\backslash in\backslash Delta$ and all $v,w\backslash in\; V$. proves that these vector space algebras are the very universal algebras that any standard vector space defines by its addition and the scalar multiplications by any given scalar (namely each $a\backslash in\; F$ gets a $\backslash delta\backslash in\backslash Delta$ such that $\backslash delta(v)=av$). proves that every such a universal algebra defines a suitable field. (Hence, it proves that these conditions imply all the axioms of the standard formal definition, as well as all the defining properties in definition 3 of a field.)〔This construction also concern the space with only one (null) vector, where the defined field only has the zero as element. This slightly extends definition 3 of a field, which only considers fields with at least two elements.〕 Since the field is defined from such vector space algebra, this is an ''algebraic'' construction of fields, which is an instance of a more general algebraic construction: the "endowed dilatation monoid" . However, as far as fields are concerned, there also is a ''geometric'' construction. Chapter II in shows how to get them starting from geometric axioms concerning points and lines. == Notes == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Vector spaces without fields」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.