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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. == Definition == Suppose that ''S'' is a set and • is some binary operation , then ''S'' with • is a monoid if it satisfies the following two axioms: ;Associativity: For all ''a'', ''b'' and ''c'' in ''S'', the equation holds. ;Identity element: There exists an element ''e'' in ''S'' such that for every element ''a'' in ''S'', the equations hold. In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.〔If both ''e''1 and ''e''2 satisfy the above equations, then ''e''1 = ''e''1 • ''e''2 = ''e''2.〕 A monoid in which each element has an inverse is a group. Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written and . This notation does not imply that it is numbers being multiplied. 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「monoid」の詳細全文を読む スポンサード リンク
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