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In mathematics, a near-semiring (also ''seminearring'') is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on semigroups. == Definition == A near-semiring is a nonempty set ''S'' with two binary operations "+" and "·", and a constant 0 such that (''S''; +; 0) is a monoid (not necessarily commutative), (''S''; ·) is a semigroup, these structures are related by one (right or left) distributive law, and accordingly the 0 is one (right or left, respectively) side absorbing element. Formally, an algebraic structure (''S''; +, ·, 0) is said to be a near-semiring if it satisfies the following axioms: # (''S''; +, 0) is a monoid, # (''S''; ·) is a semigroup, # (''a'' + ''b'') · ''c'' = ''a'' · ''c'' + ''b'' · ''c'', for all ''a'', ''b'', ''c'' in ''S'', and # 0 · ''a'' = 0 for all ''a'' in ''S''. Near-semirings are a common abstraction of semirings and near-rings (1999; Pilz, 1983 ). The standard examples of near-semirings are typically of the form ''M''(Г), the set of all mappings on a semigroup (Г; +) with identity zero, with respect to pointwise addition and composition of mappings, and certain subsets of this set. Another example are the ordinals under the usual operations of ordinal arithmetic. Strictly speaking, the class of all ordinals is not a set, so the above example should be more appropriately called a ''class near-semiring''. We get a semi-ring in the standard sense if we restrict to those ordinals which are strictly less than some multiplicatively indecomposable ordinal. 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「Near-semiring」の詳細全文を読む スポンサード リンク
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