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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 Monogenic semigroup ： ウィキペディア英語版 Monogenic semigroup In mathematics, a monogenic semigroup is a semigroup generated by a set containing only a single element. Monogenic semigroups are also called cyclic semigroups. ==Structure== The monogenic semigroup generated by the singleton set is denoted by $\backslash langle\; a\; \backslash rangle$. The set of elements of $\backslash langle\; a\; \backslash rangle$ is . There are two possibilities for the monogenic semigroup $\backslash langle\; a\; \backslash rangle$: *''a'' ''m'' = ''a'' ''n'' ⇒ ''m'' = ''n''. *There exist ''m'' ≠ ''n'' such that ''a'' ''m'' = ''a'' ''n''. In the former case $\backslash langle\; a\; \backslash rangle$ is isomorphic to the semigroup ( , + ) of natural numbers under addition. In such a case, $\backslash langle\; a\; \backslash rangle$ is an ''infinite monogenic semigroup'' and the element ''a'' is said to have ''infinite order''. It is sometimes called the ''free monogenic semigroup'' because it is also a free semigroup with one generator. In the latter case let ''m'' be the smallest positive integer such that ''a'' ''m'' = ''a'' ''x'' for some positive integer ''x'' ≠ ''m'', and let ''r'' be smallest positive integer such that ''a'' ''m'' = ''a'' ''m'' + ''r''. The positive integer ''m'' is referred to as the index and the positive integer ''r'' as the period of the monogenic semigroup $\backslash langle\; a\; \backslash rangle$. The order of ''a'' is defined as ''m''+''r''-1. The period and the index satisfy the following properties: *''a'' ''m'' = ''a'' ''m'' + ''r'' *''a'' ''m'' + ''x'' = ''a'' ''m'' + ''y'' if and only if ''m'' + ''x'' ≡ ''m'' + ''y'' ( mod ''r'' ) *$\backslash langle\; a\; \backslash rangle$ = *''K''''a'' = is a cyclic subgroup and also an ideal of $\backslash langle\; a\; \backslash rangle$. It is called the ''kernel'' of ''a'' and it is the minimal ideal of the monogenic semigroup $\backslash langle\; a\; \backslash rangle$.〔http://www.encyclopediaofmath.org/index.php/Kernel_of_a_semi-group〕〔http://www.encyclopediaofmath.org/index.php/Minimal_ideal〕 The pair ( ''m'', ''r'' ) of positive integers determine the structure of monogenic semigroups. For every pair ( ''m'', ''r'' ) of positive integers, there does exist a monogenic semigroup having index ''m'' and period ''r''. The monogenic semigroup having index ''m'' and period ''r'' is denoted by ''M'' ( ''m'', ''r'' ). The monogenic semigroup ''M'' ( 1, ''r'' ) is the cyclic group of order ''r''. The results in this section actually hold for any element ''a'' of an arbitrary semigroup and the monogenic subsemigroup $\backslash langle\; a\; \backslash rangle$ it generates. 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「Monogenic semigroup」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.