A note on triangulation of PostAlgebras and ≪leibnizian≫ lattices

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 powerset P(Ω):Δ^s(T) is then isomorphic to the postian lattice P_s+1(Ω) of the (s+1)-ordered partitions of Q, which is a multivalued genereralization of the powerset. 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-1> and also defines, with the help of triangulation, the lattices P_{s, r}(Ω) which are 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 condensations of Post multivalued logical conceptions.