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 ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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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 ...

Including:

• Complete lattice - Formal definition
• Complete lattice - Complete semilattices
• Complete lattice - Examples
• Complete lattice - Morphisms of complete lattices
• Complete lattice - Free construction and completion
• Complete lattice - Free complete semilattices
• Complete lattice - Free complete lattices
• Complete lattice - Completion
• Complete lattice - Representation
• Complete lattice - Further results
• Complete lattice - Literature

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Complete lattice - Free complete semilattices. 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 ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if and , for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more spe ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

Read more here: » Complete lattice: Encyclopedia II - Complete lattice - Morphisms of complete lattices

There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one. Thus we immediately find that every complete lattice is isomorphic to a complete lattice of sets. Another representation is obtained by noting that the image of any closure operator on a complete lattice is again a comp ...

See also:

Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

Read more here: » Complete lattice: Encyclopedia II - Complete lattice - Representation