著者

銀林 浩

出版者

明治大学

雑誌

明治大学科学技術研究所紀要 (ISSN:03864944)

巻号頁・発行日

no.9, pp.251-260, 1971-01

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Generally, it is said that the operations form a group. But this holds only because we neglect the intermediary process of the operation. So we must consider the notion of the changes which include the intermediary process, and call it pseudo-operation. We consider the totality of such pseudo-operations and its algebraic structure. By a pseudo-group, we mean a set G where a law of composition (a, b) → ab and a equivalence relation a〜b are defined and satisfy the following axioms; (1) if a〜b, then ac〜bc and ca〜cb for all c∈G, (2) weak-associative law (ab)c〜a(bc), (3) exisitence of a unit element e such that ae=ea=a for all a∈G, (4) for any a there exists a inverse element a' such that aa'〜a'a〜e. We give several examples of different kinds of pseudo-groups. Then we determine the subsystems of such an algebraic structure and show that an important subsystem is saturated pseudo-subgroup with respect to the equivalence relation. We also determine the normal subsystem such that the quotient set has also the same structure. It is showed that the quotient set of pseudo-group G with respect to the equivalence relation is a group G^^- and that saturated, pseudo-subgroups of G correspond to subgroups, of G.