Definition 3.1 (Rational terms). A rational term $f : A \to B$ between types is an expression built from the following atomic functions (Figure 1) using composition $f \mathbin{;} g$ and the functoriality combinators $f \times g$, $f + g$ and $f^*$ of products, co-products and lists:

1. the identity $A \to A$;
2. the co-projections $\mathsf{Left} : A \to A + B$ and $\mathsf{Right} : B \to A + B$;
3. the co-diagonal $A + A \to A$;
4. the projections $\mathsf{fst} : A \times B \to A$ and $\mathsf{snd} : A \times B \to B$;
5. the units $A \to A \times 1$ and $A \to 1 \times A$, and associativity of products in both directions;
6. distributivity $A \times (B + C) \to A \times B + A \times C$ and $(A + B) \times C \to A \times C + B \times C$;
7. the list constructors and destructors $\mathsf{cons} : 1 + A \times A^* \to A^*$, $\mathsf{uncons}$, $\mathsf{snoc} : 1 + A^* \times A \to A^*$ and $\mathsf{unsnoc}$;
8. list concatenation $\mathsf{concat} : A^{**} \to A^*$;
9. $\mathsf{split} : (A + B)^* \to A^* \times (B \times A^*)^*$, which cuts a list before every element of $B$;
10. bracket shifting $\mathsf{shiftR} : (A \times B)^* \times A \to A \times (B \times A)^*$ and its inverse $\mathsf{shiftL}$;
11. prefix multiplication $\mathsf{pref} : (A \times G)^* \to (A \times G)^*$ for a finite group $G$, which replaces the $i$-th group element by the product of the first $i$ ones.

The denotation $[\![ f ]\!]$ of a rational term is the function it defines. A rational term is aperiodic if it does not use prefix multiplication.