We denote by $\mathbb{S}$ the free algebra over $\mathbb{N}$ with two function symbols of arity 2, denoted $\oplus$ and $\otimes$. Elements of $\mathbb{S}$ are called *sizes*, and we'll sometimes refer to $\mathbb{S}$ as the *algebra of sizes*.

We denote by $\mathbb{H}$ the free algebra over $\ast = \{\ast\}$ with two function symbols of arity 2, denoted $\oplus$ and $\otimes$. Elements of $\mathbb{H}$ are called *shapes*, and we'll sometimes refer to $\mathbb{H}$ as the *algebra of shapes*. Define $h : \mathbb{N} \rightarrow \ast$ by $h(k) = \ast$, and let $H : \mathbb{S} \rightarrow \mathbb{H}$ be the map induced by $h$. If $s \in \mathbb{S}$, we say $H(s)$ is the *shape* of $s$.

Note that $(\mathbb{N},+,\times)$ is an algebra with two function symbols of arity 2. Let $D : \mathbb{S} \rightarrow \mathbb{N}$ be the map induced by the identity function on $\mathbb{N}$. If $s \in \mathbb{S}$, we say $D(s)$ is the *dimension* of $s$.

We denote by $\mathbb{I}$ the free algebra over $\mathbb{N}$ with two function symbols of arity 1 and one of arity 2, denoted $\mathsf{L}$, $\mathsf{R}$, and $\&$. Elements of $\mathbb{I}$ are called *indices*, and we'll sometimes refer to $\mathbb{I}$ as the *algebra of indices*.