One first defines the triangles of a lattice T, that is, the lattices Δ^s (T) of all decreasing sequences of s elements of T, and study some basic properties (modularity, distributivity, boundedness, atomisticity, inf-pseudo complementation, monotonic representations) of Δ^s (T). An important result is: If T is a boolean algebra, then Δ^s (T) is a Post algebra (of order (s+1)); one specially discusses the case when T is the powerest P(Ω) : Δ^s (T) is then isomorphic to the postian lattice P_{s+1} (Ω) of the (s+1)-ordered partitions of Ω which is a multivalued genereralization of the powerest. Afterwards, one studies some cases where T is, in turn, a Post algebra, specially T = P_r (Ω). One then exhibits some typical finite distributive lattices called leibnizians, denoted (_s^{r + s - 1} ) and also defines, with the help of triangulation, the lattices P_{s,r} (Ω) which arle called the postians of type (s,r) of a set Ω. Actually both structures (leibnizians as well as postians) turn out to be important algebraic concretisations of Post multivalued logical conceptions.