Def/Binary operation

Logical Connections

This definition logically relies on the following definitions and statements: Def/Cartesian product, Def/Rooted ordered binary tree, Def/Recursion

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Suppose that $S$ is a set, and $\cdot$ is a binary operation on $S$. Often, we are interested in repeated application of the operation $B$, e.g., expressions such as $x \cdot (y \cdot z)$.

One may consider such repeated applications, using rooted ordered binary trees as follows. Suppose that $T$ is a rooted ordered binary tree, with $n$ leaves labelled by elements of $S$. Then, we define $\prod T$ recursively as follows:

• If $n = 1$, then $T \in S$, and we define $\prod T = T \in S$.
• Suppose now that $n > 1$, and products have been defined for all rooted ordered binary trees with fewer than $n$ leaves. Then $T = (T_\ell, T_r)$, where $T_\ell$ and $T_r$ are rooted ordered binary trees, with leaves labelled by elements of $S$. Moreover, there exist positive integers $p,q$ such that $T_\ell$ has $p$ leaves, $T_r$ has $q$ leaves, and $p+q = n$; in particular, $1 \leq p < n$ and $1 \leq q < n$. It follows that $\prod T_\ell$ and $\prod T_r$ are defined. We now define:

$$\prod T = (\prod T_\ell) \cdot (\prod T_r).$$

This article has been written by: User:Marty