#HELP WITH LATTICE MATH FREE#

In this case, the direct image and inverse image maps between the power sets are upper and lower adjoints to each other, respectively.įree construction and completion Free 'complete semilattices' Īs usual, the construction of free objects depends on the chosen class of morphisms. This also yields the insight that the introduced morphisms do basically describe just two different categories of complete lattices: one with complete homomorphisms and one with meet-preserving functions (upper adjoints), dual to the one with join-preserving mappings (lower adjoints).Ī particularly important special case is for lattices of subsets P(X) and P(Y) and a function from X to Y. Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms. By the adjoint functor theorem, a monotone map between any pair of preorders preserves all joins if and only if it is a lower adjoint, and preserves all meets if and only if it is an upper adjoint.Īs such, each join-preserving morphism determines a unique upper adjoint in the inverse direction that preserves all meets. Where f is called the lower adjoint and g is called the upper adjoint. More specific complete lattices are complete Boolean algebras and complete Heyting algebras ( locales).Ī partially ordered set ( L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in ( L, ≤). Being a special instance of lattices, they are studied both in order theory and universal algebra.Ĭomplete lattices must not be confused with complete partial orders ( cpos), which constitute a strictly more general class of partially ordered sets. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.Ĭomplete lattices appear in many applications in mathematics and computer science. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum.

A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

Partially ordered set in which all subsets have both a supremum and infimum

Help with lattice math