M(i, j)

Taxoncluster representation of (the) species tree: M(i, j) = 1 iff taxon i is in the cluster j. Additional constraints on M require the cluster set to form a binary hierarchy (tree).

C(p, q, xy)

Compatibility: C(p, q, xy) = 1 exactly if the cluster pair (p, q) has the gamete xy ∈ {01, 10, 11}. 
T(a, b, c, xyz)

Rooted triple: T(a, b, c, xyz) = 1 exactly if the rooted triple with leaf set {a, b, c} and topology xyz is displayed in M. Topologies for xyz are 011, 101, and 110 and refer to the rooted triples bca, acb and abc respectively.

D(g)

tinconsistency: D(g) = 1 if the gene vertex g is tinconsistent w.r.t. a tree represented by matrix M.
