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In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally〔Głazek (2002) p.7〕 — this originated as a joke, suggesting that rigs are ri''n''gs without ''n''egative elements, similar to using ''rng'' to mean a r''i''ng without a multiplicative ''i''dentity. == Definition == A semiring is a set ''R'' equipped with two binary operations + and ·, called addition and multiplication, such that:〔Berstel & Perrin (1985), (p. 26 )〕〔Lothaire (2005) p.211〕〔Sakarovitch (2009) pp.27–28〕 # (''R'', +) is a commutative monoid with identity element 0: ## (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') ## 0 + ''a'' = ''a'' + 0 = ''a'' ## ''a'' + ''b'' = ''b'' + ''a'' # (''R'', ·) is a monoid with identity element 1: ## (''a''·''b'')·''c'' = ''a''·(''b''·''c'') ## 1·''a'' = ''a''·1 = ''a'' # Multiplication left and right distributes over addition: ## ''a''·(''b'' + ''c'') = (''a''·''b'') + (''a''·''c'') ## (''a'' + ''b'')·''c'' = (''a''·''c'') + (''b''·''c'') # Multiplication by 0 annihilates ''R'': ## 0·''a'' = ''a''·0 = 0 This last axiom is omitted from the definition of a ring: it follows from the other ring axioms. Here it does not, and it is necessary to state it in the definition. The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group. Specifically, elements in semirings do not necessarily have an inverse for the addition. The symbol · is usually omitted from the notation; that is, ''a''·''b'' is just written ''ab''. Similarly, an order of operations is accepted, according to which · is applied before +; that is, is A commutative semiring is one whose multiplication is commutative.〔Lothaire (2005) p.212〕 An idempotent semiring is one whose ''addition'' is idempotent: ''a'' + ''a'' = ''a'',〔 that is, (''R'', +, 0) is a join-semilattice with zero. There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between ''ring'' and ''semiring'' on the one hand and ''group'' and ''semigroup'' on the other hand work more smoothly. These authors often use ''rig'' for the concept defined here.〔(For an example see the definition of rig on Proofwiki.org )〕 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「Semiring」の詳細全文を読む スポンサード リンク
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