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Complete lattice
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a complete lattice is a <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> in which all subsets have both a <a href="/facts/Supremum/oXCKVopz">supremum</a> (<a href="/facts/Join_(mathematics)/cYcLhRIR">join</a>) and an <a href="/facts/Infimum/oXCKVopz">infimum</a> (<a href="/facts/Meet_(Mathematics)/cYcLhRIR">meet</a>). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a>, only <i>pairs</i> of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. </p><p>Complete lattices appear in many applications in mathematics and <a href="/facts/Computer_science/lN3pEyPz">computer science</a>. Both <a href="/facts/Order_theory/gn6fM3oJ">order theory</a> and <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a> study them as a special class of lattices. </p><p>Complete lattices must not be confused with <a href="/facts/Complete_partial_order/a855Q53A">complete partial orders</a> (CPOs), a more general class of partially ordered sets. More specific complete lattices are <a href="/facts/Complete_Boolean_algebra/7gtfjVjp">complete Boolean algebras</a> and <a href="/facts/Complete_Heyting_algebra/sNE2vnP8">complete Heyting algebras</a> (locales). </p>