Hopf theory MOC

$K$ -bimonoid

Let $K$ be a commutative ring. A $K$ -bimonoid is a bimonoid in $K 𝖬 𝗈 𝖽$ , #m/def/ralg/hopf and thus at once a $K$ -monoid and $K$ -comonoid satisfying the compatibility conditions, which in Sweedler notation read

\sum (𝑥 𝑦) (𝑥 𝑦) (1) \otimes (𝑥 𝑦) (2) = \sum (𝑥) \sum (𝑦) 𝑥 (1) 𝑦 (1) \otimes 𝑥 (2) 𝑦 (2); Δ (𝜂) = 𝜂 \otimes 𝜂; 𝜖 (𝑥 𝑦) = 𝜖 (𝑥) 𝜖 (𝑦); 𝜖 𝜂 = 1 .

See also

Biïdeal

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