Unordered tuple of colors

Definition 12.3.1. An unordered tuple of the set \(\Oo ^{\simeq }\) is a pair \((I,x)\) where \(I\) is a finite set and where \(x\colon I \to \Oo ^{\simeq }\) is a family of colors \(x_i\) indexed by \(I\). We will often denote such unordered tuples by \(\{x_i\}_{i \in I}\).

Given two unordered tuples \(\{x_i\}_{i \in I}\) and \(\{y_j\}_{j \in J}\), we can form their unordered concatenation \[ \{x_i\}_{i \in I} \sqcup \{y_j\}_{j \in J} \quad := \quad \{z_k\}_{k \in K}, \qquad \qquad K := I \sqcup J, \quad z_i := x_i, \quad z_j := y_j. \] More generally, if we are given a finite family \(\{x^j_i\}_{i \in I_j}\) of unordered tuples for \(j \in J\), then their unordered concatenation is \(\{z_k\}_{k \in K}\) with \(K := \bigsqcup _{j \in J} I_j\) and \(z_{(j,i)} := x^j_i\).