Lattice with complements
2010 Mathematics Subject Classification: Primary: 06C15 [MSN][ZBL]
A lattice $L$ with a zero 0 and a unit 1 in which for any element $a$ there is an element $b$ (called a complement of the element $a$) such that $a\lor b=1$ and $a\land b=0$. If for any $a,b\in L$ with $a\leq b$ the interval $[a,b]$ is a complemented lattice, then $L$ is called a relatively complemented lattice. Each complemented modular lattice is a relatively complemented lattice. A lattice $L$ with a zero 0 is called:
a) a partially complemented lattice if each of its intervals of the form $[0,a]$, $a\in L$, is a complemented lattice;
b) a weakly complemented lattice if for any $a,b\in L$ with $b\nleq a$ there is an element $c\in L$ such that $a\land c=0$ and $b\land c\neq0$;
c) a semi-complemented lattice if for any $a\in L$, $a\neq1$, there is an element $b\in L$, $b\neq0$, such that $a\land b=0$;
d) a pseudo-complemented lattice if for any $a\in L$ there is an element $a^*$ such that $a\land x=0$ if and only if $x\leq a^*$; and
e) a quasi-complemented lattice if for any $x\in L$ there is an element $y\in L$ such that $x\land y=0$ and $x\lor y$ is a dense element.
Ortho-complemented lattices also play an important role (see Orthomodular lattice). See  for the relation between the various types of complements in lattices.
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In a distributive lattice, each element has at most one complement; conversely, a lattice in which each element has at most one relative complement in each interval in which it lies must be distributive.
A Boolean lattice is a complemented distributive lattice.
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Lattice with complements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice_with_complements&oldid=37425