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are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). The ortholattices are most often used in
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A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case
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in
Boolean lattices. This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.
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every element will have at most one complement. A lattice in which every element has exactly one complement is called a
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1976:
919:
443:
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140:, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a
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1971:
1607:
934:
Grätzer (1971), Lemma I.6.2, p. 48. This result holds more generally for modular lattices, see
Exercise 4, p. 50.
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A distributive lattice is complemented if and only if it is bounded and relatively complemented. The lattice of
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2008:
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217:. In other words, a relatively complemented lattice is characterized by the property that for every element
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in quantum logic is "formally indistinguishable from the calculus of linear subspaces with respect to
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operation, provides an example of an orthocomplemented lattice that is not, in general, distributive.
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A lattice with the property that every interval (viewed as a sublattice) is complemented is called a
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is therefore defined as an orthocomplemented lattice such that for any two elements the implication
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and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the
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The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented.
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914:, Encyclopedia of Mathematics and its Applications, Cambridge University Press, p. 29,
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is a bounded lattice equipped with an orthocomplementation. The lattice of subspaces of an
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provide an example of a complemented lattice that is not, in general, distributive.
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The
Unapologetic Mathematician: Orthogonal Complements and the Lattice of Subspaces
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In general an element may have more than one complement. However, in a (bounded)
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there are various competing definitions of "Orthocomplementation" in literature.
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represent quantum propositions and behave as an orthocomplemented lattice.
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110:, viewed as a bounded lattice in its own right, is a complemented lattice.
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Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25.
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323: in this section. Unsourced material may be challenged and removed.
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Lattices of this form are of crucial importance for the study of
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Orthocomplemented lattices, like
Boolean algebras, satisfy
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and orthogonal complements" corresponding to the roles of
460:
on a bounded lattice is a function that maps each element
1022:
Lattice Theory: First
Concepts and Distributive Lattices
468:
in such a way that the following axioms are satisfied:
189:= 1 and
567:, the node on the right-hand side has two complements.
970:
242: and
147:
824:, since they are part of the axiomisation of the
2176:
977:Ranganathan Padmanabhan; Sergiu Rudeanu (2008).
99: = 0. Complements need not be unique.
1494:
1398:
911:Semimodular Lattices: Theory and Applications
2152:Positive cone of a partially ordered group
1501:
1487:
1405:
1391:
1057:
444:Learn how and when to remove this message
383:Learn how and when to remove this message
2135:Positive cone of an ordered vector space
1007:
980:Axioms for lattices and boolean algebras
692:
15:
1038:
1016:
943:Birkhoff (1961), Corollary IX.1, p. 134
282:
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1482:
952:
907:
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321:adding citations to reliable sources
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221:in an interval there is an element
767:is modular, but not distributive.
23:of a complemented lattice. A point
13:
1662:Properties & Types (
1043:. Basel, Switzerland: Birkhäuser.
14:
2196:
2118:Positive cone of an ordered field
1067:
1058:Rutherford, Daniel Edwin (1965).
983:. World Scientific. p. 128.
1972:Ordered topological vector space
1508:
1012:. American Mathematical Society.
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591:
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117:on a complemented lattice is an
1452:"Uniquely complemented lattice"
760:; e.g. the above-shown lattice
605:admits 3 orthocomplementations.
586:admits no orthocomplementation.
308:needs additional citations for
215:relatively complemented lattice
148:Definition and basic properties
104:relatively complemented lattice
35:are complements if and only if
1060:Introduction to Lattice Theory
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946:
937:
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901:
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1:
1929:Series-parallel partial order
1024:. W. H. Freeman and Company.
1001:
209:uniquely complemented lattice
106:is a lattice such that every
1608:Cantor's isomorphism theorem
7:
1648:Szpilrajn extension theorem
1623:Hausdorff maximal principle
1598:Boolean prime ideal theorem
1466:"Orthocomplemented lattice"
873:
756:holds. This is weaker than
419:. The specific problem is:
268:relative to the interval.
164:1), in which every element
156:is a bounded lattice (with
71:1), in which every element
10:
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1994:Topological vector lattice
1008:Birkhoff, Garrett (1961).
880:Pseudocomplemented lattice
549:Some complemented lattices
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264:is called a complement of
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1952:
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1590:
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1516:
531:orthocomplemented lattice
1603:CantorâBernstein theorem
1039:Grätzer, George (1978).
885:
560:In the pentagon lattice
464:to an "orthocomplement"
2147:Partially ordered group
1967:Specialization preorder
908:Stern, Manfred (1999),
1633:Kruskal's tree theorem
1628:KnasterâTarski theorem
1618:DushnikâMiller theorem
1424:"Complemented lattice"
1041:General Lattice Theory
332:"Complemented lattice"
44:
1438:"Relative complement"
693:Orthomodular lattices
543:orthogonal complement
138:distributive lattices
19:
2125:Ordered vector space
1302:Group with operators
1245:Complemented lattice
1080:Algebraic structures
780:orthomodular lattice
701:if for all elements
697:A lattice is called
579:The diamond lattice
458:orthocomplementation
426:improve this article
415:to meet Knowledge's
317:improve this article
283:Orthocomplementation
205:distributive lattice
154:complemented lattice
131:orthomodular lattice
115:orthocomplementation
57:complemented lattice
1963:Alexandrov topology
1909:Lexicographic order
1868:Well-quasi-ordering
1356:Composition algebra
1116:Quasigroup and loop
539:inner product space
91: = 1 and
1944:Transitive closure
1904:Converse/Transpose
1613:Dilworth's theorem
1062:. Oliver and Boyd.
843:observed that the
172:, i.e. an element
79:, i.e. an element
45:
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2130:Partially ordered
1939:Symmetric closure
1924:Reflexive closure
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1031:978-0-7167-0442-3
990:978-981-283-454-6
833:quantum mechanics
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408:This article may
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1111:Rack and quandle
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713:the implication
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1685:Boolean algebra
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1638:Laver's theorem
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1547:Binary relation
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1268:Map of lattices
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1777:Join and meet
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1675:Antisymmetric
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1582:Weak ordering
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1567:Partial order
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955:, p. 11.
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846:
845:propositional
842:
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827:
826:Hilbert space
823:
822:quantum logic
818:
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641:Hilbert space
639:
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627:quantum logic
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334: â
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328:Find sources:
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306:This section
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158:least element
155:
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129:is called an
128:
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105:
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98:
95: â§
94:
90:
87: â¨
86:
82:
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70:
66:
65:least element
62:
59:is a bounded
58:
54:
50:
34:
22:
21:Hasse diagram
18:
1956:& Orders
1934:Star product
1863:Well-founded
1816:Prefix order
1772:Distributive
1762:Complemented
1761:
1732:Foundational
1697:Completeness
1653:Zorn's lemma
1557:Cyclic order
1540:Key concepts
1510:Order theory
1469:
1455:
1441:
1427:
1376:Hopf algebra
1314:
1307:Vector space
1272:
1244:
1212:
1141:Group theory
1139:
1104: /
1059:
1040:
1021:
1009:
979:
972:
960:
953:Stern (1999)
948:
939:
930:
910:
903:
894:
867:
863:
859:
852:set products
819:
816:
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685:
681:
677:
673:
667:
663:
659:
655:
645:
629:, where the
621:
599:
598:The lattice
580:
561:
535:ortholattice
534:
530:
528:
522:
518:
514:
510:
500:
496:
487:
483:
479:
475:
465:
461:
457:
455:
440:
431:
424:Please help
420:
409:
379:
370:
360:
353:
346:
339:
327:
315:Please help
310:verification
307:
277:vector space
270:
265:
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56:
53:order theory
49:mathematical
46:
2140:Riesz space
2101:Isomorphism
1977:Normal cone
1899:Composition
1833:Semilattice
1742:Homogeneous
1727:Equivalence
1577:Total order
1361:Lie algebra
1346:Associative
1250:Total order
1240:Semilattice
1214:Ring theory
856:linear sums
829:formulation
434:August 2014
428:if you can.
373:August 2014
136:In bounded
127:modular law
83:satisfying
27:and a line
2108:Order type
2042:Cofinality
1883:Well-order
1858:Transitive
1747:Idempotent
1680:Asymmetric
1471:PlanetMath
1457:PlanetMath
1443:PlanetMath
1429:PlanetMath
1002:References
541:, and the
343:newspapers
287:See also:
225:such that
176:such that
170:complement
119:involution
77:complement
33:Fano plane
2159:Upper set
2096:Embedding
2032:Antichain
1853:Tolerance
1843:Symmetric
1838:Semiorder
1784:Reflexive
1702:Connected
1371:Bialgebra
1177:Near-ring
1134:Lie group
1102:Semigroup
638:separable
634:subspaces
273:subspaces
2179:Category
1954:Topology
1821:Preorder
1804:Eulerian
1767:Complete
1717:Directed
1707:Covering
1572:Preorder
1531:Category
1526:Glossary
1207:Lie ring
1172:Semiring
1020:(1971).
874:See also
848:calculus
482:= 1 and
410:require
121:that is
108:interval
2059:Duality
2037:Cofinal
2025:Related
2004:FrĂŠchet
1881:)
1757:Bounded
1752:Lattice
1725:)
1723:Partial
1591:Results
1562:Lattice
1338:Algebra
1330:Algebra
1235:Lattice
1226:Lattice
817:holds.
796:, then
727:, then
699:modular
412:cleanup
357:scholar
61:lattice
47:In the
31:of the
2084:Subnet
2064:Filter
2014:Normed
1999:Banach
1965:&
1872:Better
1809:Strict
1799:Graded
1690:topics
1521:Topics
1366:Graded
1297:Module
1288:Module
1187:Domain
1106:Monoid
1047:
1028:
987:
918:
631:closed
359:
352:
345:
338:
330:
168:has a
160:0 and
75:has a
67:0 and
63:(with
2074:Ideal
2052:Graph
1848:Total
1826:Total
1712:Dense
1332:-like
1290:-like
1228:-like
1197:Field
1155:-like
1129:Magma
1097:Group
1091:-like
1089:Group
886:Notes
778:. An
739:) = (
636:of a
517:then
364:JSTOR
350:books
275:of a
1665:list
1162:Ring
1153:Ring
1045:ISBN
1026:ISBN
985:ISBN
916:ISBN
866:and
839:and
808:) =
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709:and
680:) =
662:) =
490:= 0.
336:news
197:= 0.
55:, a
2079:Net
1879:Pre
1167:Rng
868:not
860:and
831:of
800:⨠(
788:if
731:⨠(
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533:or
529:An
509:if
456:An
319:by
113:An
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864:or
862:,
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806:c
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81:b
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41:l
37:p
29:l
25:p
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