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Definitions<br />
A monoid is a triple (A, ⊗, 1) where<br />
1. ⊗ is a closed associative binary operator on the set A,<br />
2. 1istheidentity element for ⊗, i.e., for all a ∈ A, a ⊗ 1=1 ⊗ a = a.<br />
A monoid is commutative if ⊗ is commutative.<br />
A semiring is a 5-tuple R =(A, ⊕, ⊗, 0, 1) such that<br />
1. (A, ⊕, 0) is a commutative monoid.<br />
2. (A, ⊗, 1) is a monoid.<br />
3. ⊗ distributes over ⊕: for all a, b, c in A,<br />
(a ⊕ b) ⊗ c =(a ⊗ c) ⊕ (b ⊗ c),<br />
c ⊗ (a ⊕ b) =(c ⊗ a) ⊕ (c ⊗ b).<br />
([0, 1], +, 0)<br />
([0, 1], , 1)<br />
([0, 1], max, 0)<br />
([0, 1], max, , 0, 1)<br />
([0, 1], +, , 0, 1)<br />
4. 0isanannihilator for ⊗: for all a in A, 0 ⊗ a = a ⊗ 0=0.<br />
Liang Huang<br />
7<br />
Dynamic Programming