Jónsson–Tarski algebra

Example

A Jónsson–Tarski algebra is an algebra $(A, \cdot, L, R)$ with a binary operation $\cdot$ and two unary operations $L$ and $R$ such that, for all $x, y \in A$:

1. $L(x \cdot y) = x$;
2. $R(x \cdot y) = y$;
3. $L(x) \cdot R(x) = x$.

These laws say exactly that $(x, y) \mapsto x \cdot y$ is a bijection from $A \times A$ to $A$, with inverse $x \mapsto (L(x), R(x))$. Since the laws are identities, Jónsson–Tarski algebras form a variety.

A concrete instance is the set $A$ of infinite binary sequences $s = (s_0, s_1, s_2, \dots)$. Let $s \cdot t = (s_0, t_0, s_1, t_1, \dots)$ interleave the two sequences. Let $L(s) = (s_0, s_2, s_4, \dots)$ and $R(s) = (s_1, s_3, s_5, \dots)$. Then $L(s \cdot t) = s$ and $R(s \cdot t) = t$ because the even places of $s \cdot t$ hold $s$ and the odd places hold $t$. Also $L(s) \cdot R(s) = s$ because interleaving the even and odd places rebuilds $s$.