Definition 2.7.2. We define:
SemiRing :=: {[R, ⊕, 0, ⊙, 1] | R is a set, ⊕ : R × R → R is an operation on R, 0 ∈ R, ⊙ : R × R → R is an operation on R, 1 ∈ R such that (R, ⊕, 0, ⊙, 1) forms a semiring} [R, ⊕, 0, ⊙, 1] = [S, ⋆, 0, ∗, 1] :⇔ ∃ φ : R ↔ S : [⊕ ≃φ ⋆ and 0 ≃φ 0 and ⊙ ≃φ ∗ and 1 ≃φ 1] (R is a set, ⊕ : R × R → R is an operation on R, 0 ∈ R, ⊙ : R × R → R is an operation on R, 1 ∈ R, S is a set, ⋆ : S × S → S is an operation on S, 0 ∈ S, ∗ : S × S → S is an operation on S, 1 ∈ S with suitable conditions) We write “let R be a semiring” for “let R ∈ SemiRing.”
References.