View source for Lattice with complements

{{TEX|done}}{{MSC|06C15}}

''complemented lattice''

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 [[#References|[4]]] for the relation between the various types of complements in lattices.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,   "Elements of lattice theory" , A. Hilger  (1977)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Skornyakov,   "Complemented modular lattices and regular rings" , Oliver &amp; Boyd  (1964)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  P.A. Grillet,   J.C. Varlet,   "Complementedness conditions in lattices"  ''Bull Soc. Roy. Sci. Liège'' , '''36''' :  11–12  (1967)  pp. 628–642</TD></TR>
</table>



====Comments====
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.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Beran,   "Orthomodular lattices" , Reidel  (1985)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Grätzer,   "Lattice theory" , Freeman  (1971)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M.L. Dubreil-Jacotin,   L. Lesieur,   R. Croiset,   "Leçons sur la théorie des treillis" , Gauthier-Villars  (1953)</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) {{ISBN|978-0-521-78451-1}} {{ZBL|1002.06001}}</TD></TR>
</table>