User:Dfeuer/Transitive Closure of Relation Compatible with Operation is Compatible

Theorem
Let $(S, \circ)$ be a magma.

Let $R$ be a relation compatible with $\circ$.

Let $T$ be the transitive closure of $R$.

Then $T$ is compatible with $\circ$.

Proof
Let $a, b, c \in S$.

Let $a \mathrel{T} b$.

Then for some $n$, there is a finite sequence $x_0, \dots, x_n$ in $S$ such that:
 * $x_0 = a$
 * $x_n = b$
 * For each $k = 0, \dots, n-1$: $x_k \mathrel{R} x_{k+1}$

Since $R$ is compatible with $\circ$, for each $k = 0, \dots, n-1$:
 * $c \circ x_k \mathrel{R} c \circ x_{k+1}$

Thus by the definition of transitive closure:
 * $c \circ a \mathrel{R} c \circ b$

A similar argument shows that:
 * $a \circ c \mathrel{R} b \circ c$

Thus $T$ is compatible with $\circ$.