Lattice (order)

A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.

The algebraic interpretation of lattices plays an essential role in universal algebra.

Further examples of lattices are given for each of the additional properties discussed below.

Most partially ordered sets are not lattices, including the following.

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.

Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.

Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.

Another equivalent (for graded lattices) condition is Birkhoff's condition:

Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via Scott information systems.

Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.

Elementary texts recommended for those with limited mathematical maturity:

The standard contemporary introductory text, somewhat harder than the above: