Complete lattice

In order theory, arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see completeness (order theory)). In effect, this means that it is sufficient to require the existence of either all meets or all joins to obtain the class of all complete lattices.

As a consequence, some authors use the terms complete meet-semilattice or complete join-semilattice as another way to refer to complete lattices. Though similar on objects, the terms entail different notions of homomorphism, as will be explained in the below section on morphisms.

On the other hand, some authors have no use for this distinction of morphisms (especially since the emerging concepts of "complete semilattice morphisms" can as well be specified in general terms). Consequently, complete meet-semilattices have also been defined as those meet-semilattices that are also complete partial orders. This concept is arguably the "most complete" notion of a meet-semilattice that is not yet a lattice (in fact, only the top element may be missing). This discussion is also found in the article on semilattices.

A sublattice M of a complete lattice L is called a complete sublattice of L if for every subset A of M the elements ${\displaystyle \bigwedge A}$ and ${\displaystyle \bigvee A}$, as defined in L, are actually in M.[1]

If the above requirement is lessened to require only non-empty meet and joins to be in M, the sublattice M is called a closed sublattice of L.

A lattice is said to be "conditionally complete" if it satisfies either or both of the following properties:[2]

• Any subset bounded above has a least upper bound
• Any subset bounded below has a greatest lower bound

Furthermore, morphisms that preserve all joins are equivalently characterized as the lower adjoint part of a unique Galois connection. For any pair of preorders P and Q, these are given by pairs of monotone functions f and g such that

${\displaystyle f(x)\leq y\iff x\leq g(y)}$

where f is called the lower adjoint and g is called the upper adjoint. 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.

As such, each join-preserving morphism determines a unique upper adjoint in the inverse direction that preserves all meets. Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as 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).

A particularly important special case is for lattices of subsets P(X) and P(Y) and a function from X to Y. In this case, the direct image and inverse image maps between the power sets are upper and lower adjoints to each other, respectively.

As usual, the construction of free objects depends on the chosen class of morphisms. Let us first consider functions that preserve all joins (i.e. lower adjoints of Galois connections), since this case is simpler than the situation for complete homomorphisms. Using the aforementioned terminology, this could be called a free complete join-semilattice.

Using the standard definition from universal algebra, a free complete lattice over a generating set S is a complete lattice L together with a function i:S→L, such that any function f from S to the underlying set of some complete lattice M can be factored uniquely through a morphism f° from L to M. Stated differently, for every element s of S we find that f(s) = f°(i(s)) and that f° is the only morphism with this property. These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of complete lattices and join-preserving functions which is left adjoint to the forgetful functor from complete lattices to their underlying sets.

Free complete lattices in this sense can be constructed very easily: the complete lattice generated by some set S is just the powerset 2^S, i.e. the set of all subsets of S, ordered by subset inclusion. The required unit i:S→2^S maps any element s of S to the singleton set {s}. Given a mapping f as above, the function f°:2^S→M is defined by

${\displaystyle f^{\circ }(X)=\bigvee \{f(s)|s\in X\}}$.

Then f° transforms unions into suprema and thus preserves joins.

Our considerations also yield a free construction for morphisms that do preserve meets instead of joins (i.e. upper adjoints of Galois connections). In fact, we merely have to dualize what was said above: free objects are given as powersets ordered by reverse inclusion, such that set union provides the meet operation, and the function f° is defined in terms of meets instead of joins. The result of this construction could be called a free complete meet-semilattice. One should also note how these free constructions extend those that are used to obtain free semilattices, where we only need to consider finite sets.

The situation for complete lattices with complete homomorphisms obviously is more intricate. In fact, free complete lattices generally do not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument-sets of every cardinality.

This property in itself is not a problem: as the case of free complete semilattices above shows, it can well be that the solution of the word problem leaves only a set of equivalence classes. In other words, it is possible that proper classes of the class of all terms have the same meaning and are thus identified in the free construction. However, the equivalence classes for the word problem of complete lattices are "too small", such that the free complete lattice would still be a proper class, which is not allowed.

Now one might still hope that there are some useful cases where the set of generators is sufficiently small for a free complete lattice to exist. Unfortunately, the size limit is very low and we have the following theorem:

The free complete lattice on three generators does not exist; it is a proper class.

A proof of this statement is given by Johnstone;[3] the original argument is attributed to Alfred W. Hales;[4] see also the article on free lattices.

If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction.

As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion. For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above.

The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set. Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above.