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). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets. More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales). A 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, ≤). The meet is denoted by ⋀ A {displaystyle igwedge A} , and the join by ⋁ A {displaystyle igvee A} . 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 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. 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.