Following Aguiló-Suñer-Torrens (2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we give characterizations of meet-irreducible discrete copulas from an order-theoretical aspect: we show that the set of all irreducible discrete copulas is a lattice in analogy with Nelsen and Úbeda-Flores (2005). Moreover, we clarify its lattice structure related to Kendall's $τ$ and Spearman's $ρ$ borrowing ideas from Coxeter groups.