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Complete lattice - Formal definition

Complete lattice - Formal definition: Encyclopedia II - Complete lattice - Formal definition

A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least upper bound (supremum, join). These are denoted by: A (meet) and A (join). Note that in the special case where A is the empty set the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete latt ...

Complete lattice: Encyclopedia II - Complete lattice - Formal definition

Complete lattice - Formal definition

A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least upper bound (supremum, join). These are denoted by:

A (meet) and A (join).

Note that in the special case where A is the empty set the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete lattices do thus form a special class of bounded lattices.

More implications of the above definition are discussed in the article on completeness properties in order theory.

Complete lattice - Complete semilattices

It is a well-known fact of order theory that 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 homomorphisms, 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.