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''complemented lattice''
 
''complemented lattice''
  
A [[Lattice|lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576801.png" /> with a zero 0 and a unit 1 in which for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576802.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576803.png" /> (called a complement of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576804.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576806.png" />. If for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576807.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576808.png" /> the [[Interval|interval]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l0576809.png" /> is a complemented lattice, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768010.png" /> is called a relatively complemented lattice. Each complemented [[Modular lattice|modular lattice]] is a relatively complemented lattice. A lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768011.png" /> with a zero 0 is called: a) a partially complemented lattice if each of its intervals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768013.png" />, is a complemented lattice; b) a weakly complemented lattice if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768015.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768018.png" />; c) a semi-complemented lattice if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768020.png" />, there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768022.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768023.png" />; d) a pseudo-complemented lattice if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768024.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768026.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768027.png" />; and e) a quasi-complemented lattice if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768028.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057680/l05768031.png" /> is a dense element. Ortho-complemented lattices also play an important role (see [[Orthomodular lattice|Orthomodular lattice]]). See [[#References|[4]]] for the relation between the various types of complements in lattices.
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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:
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a) a ''partially complemented'' lattice if each of its intervals of the form $[0,a]$, $a\in L$, is a complemented lattice;  
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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 $b\land c\neq0$;  
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c) a ''semi-complemented'' lattice if for any $a\in L$, $a\neq1$, there is an element $b\in L$, $b\neq0$, such that $a\land b=0$;  
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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  
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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.  
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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====
 
====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , A. Hilger  (1977)  (Translated from Russian)</TD></TR>
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<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>
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<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>
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</table>
  
  
  
 
====Comments====
 
====Comments====
In a [[Distributive lattice|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.
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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.
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A ''[[Boolean lattice]]'' is a complemented distributive lattice.
  
 
====References====
 
====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 treilles" , Gauthier-Villars  (1953)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Beran,  "Orthomodular lattices" , Reidel  (1985)</TD></TR>
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<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>
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<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>
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</table>

Latest revision as of 08:09, 26 November 2023

2020 Mathematics Subject Classification: Primary: 06C15 [MSN][ZBL]

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

References

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


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

[a1] L. Beran, "Orthomodular lattices" , Reidel (1985)
[a2] G. Grätzer, "Lattice theory" , Freeman (1971)
[a3] M.L. Dubreil-Jacotin, L. Lesieur, R. Croiset, "Leçons sur la théorie des treillis" , Gauthier-Villars (1953)
[b1] 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
How to Cite This Entry:
Lattice with complements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice_with_complements&oldid=12763
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article