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Subset Product (Algebra)**

Given an algebra \((\mathcal{A},\diamond)\) over a field \(k\) and two subsets \(S,T\subseteq\mathcal{A}\), their product \(S\diamond T\) is the subspace of \(\mathcal{A}\) spanned by all pairs \(s\diamond t\) with \(s\in S,\:t\in T\), i.e. \(S\diamond T:=\left\{\sum_{i=1}^na_i(s_i\diamond t_i)\middle|a_i\in k,\:s_i\in S,\:t_i\in T\right\}\).