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Pseudocomplement
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, a pseudocomplement is one generalization of the notion of <a href="/facts/Complemented_lattice/KpwmsACp">complement</a>. In a <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> <i>L</i> with <a href="/facts/Bottom_element/JVn2B3mo">bottom element</a> 0, an element <i>x</i> ∈ <i>L</i> is said to have a <i>pseudocomplement</i> if there exists a <a href="/facts/Greatest_element/JVn2B3mo">greatest element</a> <i>x</i>* ∈ <i>L</i> with the property that <i>x</i> ∧ <i>x</i>* = 0. More formally, <i>x</i>* = max{ <i>y</i> ∈ <i>L</i> | <i>x</i> ∧ <i>y</i> = 0 }. The lattice <i>L</i> itself is called a pseudocomplemented lattice if every element of <i>L</i> is pseudocomplemented. Every pseudocomplemented lattice is necessarily <a href="/facts/Bounded_set/yCldjtoy">bounded</a>, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a <i>p</i>-algebra. However this latter term may have other meanings in other areas of mathematics. </p>