$\mu, \nu\in P$ with $x=\mu\nu^{-1}$  such that $\gamma, \delta\in P$ and $\gamma\delta^{-1}=\mu\nu^{-1}$ imply $\mu\leq \gamma$ and $\nu\leq \delta$. (The pair $\mu, \nu$ is unique.)}$$

Let $(G, P)$ be a quasi-lattice ordered group, and let $x, y\in G$. If $x$ and $y$ have a common upper bound in $P$, then their least common upper bound in $P$ is denoted $x\vee y$. We write $x\vee y=\infty$ when $x$ and $y$ have no common upper bound in $P$ and $x\vee y<\infty$ when they have a common upper bound. An *isometric representation of $P$* in a $C^*$-algebra $A$ is a map $W:P\to A$ such that $W_e=1$, $W_p$ is an isometry and $W_pW_q=W_{pq}$ for all $p,q\in P$. We say that $W$ is *covariant* if $$\label{covariance condition}
W_pW^*_pW_qW^*_q=\begin{cases}W_{p\vee q}W^*_{p\vee q}&\text{if $p\vee q<\infty$}\\0&\text{otherwise.}
\end{cases}$$ Equivalently, $W$ is covariant if $$W^*_pW_q=\begin{cases}W_{p^{-1}(p\vee q)}W^*_{q^{-1}(p\vee q)}&\text{if $p\vee q<\infty$}\\0&\text{otherwise.}\end{cases}$$ An example of a covariant representation is $T:P\to B(\ell^2(P))$ characterised by $T_p\epsilon_x=\epsilon_{px}$ where $\{\epsilon_x:x\in P\}$ is the orthonormal basis of point masses in $\ell^2(P)$.

In [@Nica1992 §§2.4 and 4.1] Nica examined two $C^*$-algebras associated to $(G, P)$. The reduced $C^*$-algebra $C^*_r(G,P)$ of $(G,P)$ is the $C^*$-subalgebra of $B(\ell^2(P))$ generated by $\{T_p:p\in P\}$. The universal $C^*$-algebra $C^*(G,P)$ of $(G,P)$ is generated by a universal 