$𝕂$ -submonoid

A $𝕂$ -submonoid $𝐵$ of an $𝕂$ -monoid $𝐴$ is a subalgebra $𝐵 \leq 𝐴$ containing the identity, #m/def/ralg i.e. $𝐵$ is itself an associative algebra; $𝐵 \cdot 𝐵 \subseteq 𝐵$ and $𝟙 \in 𝐵$ .