Puulattice

Puulattice is an algebraic structure in order theory. Formally, it is a tuple (P, ≤, ∧, ∨, p) where (P, ≤, ∧, ∨) is a lattice and p: P → P is a unary operation called the puu-operator or projection. The operator p is idempotent (p(p(x)) = p(x) for all x) and monotone (x ≤ y implies p(x) ≤ p(y)). The image Pp = { p(x) | x ∈ P } is required to be a sublattice of P, and p acts as a retraction onto Pp, meaning p(x) ∈ Pp and p(y) = y for all y ∈ Pp.

In many cases, puulattices arise from lattices equipped with a distinguished sublattice Q ⊆ P, with p being a projection onto Q. For instance, if P is a direct product A × B, a standard projection p((a, b)) = (a, top_B) is an example whose image is isomorphic to A and forms a sublattice of P. More generally, puulattices can model situations where a lattice is decomposed into a substructure and its complement, with p selecting the substructure component.

Puulattices are studied to understand how sublattice structure interacts with the ambient lattice, and they connect to concepts such as closure and interior operators, lattice homomorphisms, and decomposition theorems. They appear in areas including universal algebra, category theory, and theoretical computer science, where notions of projection and substructure play a key role in semantic modeling and analysis. See also lattice, projection, and closure operator.