Labelled classes, labelled product, and the constructions $\mathrm{SEQ}$, $\mathrm{SET}$, $\mathrm{CYC}$, and boxed product

A labelled class $\mathcal{A}$ assigns to each finite label set $L$ a finite set $\mathcal{A}[L]$ of $\mathcal{A}$-structures carried by $L$, together with transport along bijections of label sets. Write

$a_n:=\mathrm{\mid }\mathcal{A}[[n]]\mathrm{\mid },$

so the exponential generating function of $\mathcal{A}$ is $A(x)={\sum }_{n\ge 0}{a}_n{x}^n\mathrm{/}n!$.

If $\mathcal{A}$ and $\mathcal{B}$ are labelled classes, their labelled product $\mathcal{A}\star \mathcal{B}$ on a label set $L$ consists of triples $(S,\alpha ,\beta )$ where $S\subseteq L$, $\alpha \in \mathcal{A}[S]$, and $\beta \in \mathcal{B}[L\setminus S]$. Thus the labels are split into two disjoint parts, one carrying the $\mathcal{A}$-object and the other the $\mathcal{B}$-object.

For a labelled class $\mathcal{A}$:

• $\mathrm{SEQ}(\mathcal{A})$ is the class of finite ordered lists of pairwise disjoint $\mathcal{A}$-objects whose label sets partition the ambient label set;
• $\mathrm{SET}(\mathcal{A})$ is the class of finite unordered sets of pairwise disjoint $\mathcal{A}$-objects whose label sets partition the ambient label set;
• $\mathrm{CYC}(\mathcal{A})$ is the class of finite cyclic arrangements of pairwise disjoint $\mathcal{A}$-objects whose label sets partition the ambient label set; and
• the boxed product ${\mathcal{A}}^{\mathrm{\square }}\star \mathcal{B}$ is the subclass of $\mathcal{A}\star \mathcal{B}$ in which the smallest label belongs to the $\mathcal{A}$-part.

All four constructions keep the source convention that labels are distinct and their union is the ambient finite label set.

As with the ordinary sequence construction, these formal labelled constructions need not define labelled classes in the finiteness sense when $\mathcal{A}[\mathrm{\varnothing }]$ is nonempty. The translation theorem below therefore imposes the zero-constant-coefficient hypothesis exactly where it is needed.