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magma in nLab

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A binary operation on a set 𝑆 is a function ( − ) ⋅ ( − ) : 𝑆 × 𝑆 → 𝑆 from the Cartesian product 𝑆 × 𝑆 to 𝑆 . A magma (or binary algebraic structure, or, alternatively, a mono-binary algebra) ( 𝑆 , ⋅ ) is a set equipped with a binary operation on it. A magma is called unital if it has a neutral element; that is, an element 1 ∈ 𝑆 such that 1 ⋅ 𝑥 = 𝑥 = 𝑥 ⋅ 1 . Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1). One can consider one-sided unital elements separately: 1 𝐿 ⋅ 𝑥 = 𝑥 and/or 𝑥 = 𝑥 ⋅ 1 𝑅 . Note that units may be far from unique. commutative if the binary operation takes the same value when its two arguments are interchanged: 𝑥 ⋅ 𝑦 = 𝑦 ⋅ 𝑥 . associative if the binary operation satisfies the associativity condition ( 𝑥 ⋅ 𝑦 ) ⋅ 𝑧 = 𝑥 ⋅ ( 𝑦 ⋅ 𝑧 ) . invertible if it has an inverse element. an absorption magma if it has an element 0 ∈ 𝑆 such that the binary operation satisfies the absorption condition:

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