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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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151:, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a
2004:
1504:
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1977:
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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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228:. 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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925:, 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.
17:
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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.
121:, viewed as a bounded lattice in its own right, is a complemented lattice.
1964:
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1582:
1182:
909:
Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25.
334: in this section. Unsourced material may be challenged and removed.
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26:
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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
471:
on a bounded lattice is a function that maps each element
1033:
Lattice Theory: First
Concepts and Distributive Lattices
479:
in such a way that the following axioms are satisfied:
200:= 1 and
578:, the node on the right-hand side has two complements.
2035:
1963:
1902:
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253: and
835:, since they are part of the axiomisation of the
988:Ranganathan Padmanabhan; Sergiu Rudeanu (2008).
110: = 0. Complements need not be unique.
1505:
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922:Semimodular Lattices: Theory and Applications
8:
2163:Positive cone of a partially ordered group
1512:
1498:
1490:
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1402:
1086:
455:Learn how and when to remove this message
394:Learn how and when to remove this message
2146:Positive cone of an ordered vector space
991:Axioms for lattices and boolean algebras
1089:
954:Birkhoff (1961), Corollary IX.1, p. 134
902:
558:
963:
7:
332:adding citations to reliable sources
232:in an interval there is an element
778:is modular, but not distributive.
34:of a complemented lattice. A point
1673:Properties & Types (
1054:. Basel, Switzerland: Birkhäuser.
25:
2129:Positive cone of an ordered field
1069:Rutherford, Daniel Edwin (1965).
994:. World Scientific. p. 128.
1983:Ordered topological vector space
1023:. American Mathematical Society.
621:
602:
583:
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410:
308:
128:on a complemented lattice is an
1463:"Uniquely complemented lattice"
771:; e.g. the above-shown lattice
616:admits 3 orthocomplementations.
597:admits no orthocomplementation.
319:needs additional citations for
226:relatively complemented lattice
159:Definition and basic properties
115:relatively complemented lattice
46:are complements if and only if
1071:Introduction to Lattice Theory
1:
1940:Series-parallel partial order
1035:. W. H. Freeman and Company.
220:uniquely complemented lattice
117:is a lattice such that every
1619:Cantor's isomorphism theorem
1659:Szpilrajn extension theorem
1634:Hausdorff maximal principle
1609:Boolean prime ideal theorem
1477:"Orthocomplemented lattice"
767:holds. This is weaker than
430:. The specific problem is:
279:relative to the interval.
175:1), in which every element
167:is a bounded lattice (with
82:1), in which every element
2212:
2005:Topological vector lattice
1019:Birkhoff, Garrett (1961).
891:Pseudocomplemented lattice
560:Some complemented lattices
297:
275:is called a complement of
1527:
542:orthocomplemented lattice
1614:Cantor–Bernstein theorem
1050:Grätzer, George (1978).
571:In the pentagon lattice
475:to an "orthocomplement"
2158:Partially ordered group
1978:Specialization preorder
919:Stern, Manfred (1999),
1644:Kruskal's tree theorem
1639:Knaster–Tarski theorem
1629:Dushnik–Miller theorem
1435:"Complemented lattice"
1052:General Lattice Theory
343:"Complemented lattice"
55:
1449:"Relative complement"
704:Orthomodular lattices
554:orthogonal complement
149:distributive lattices
30:
2136:Ordered vector space
1313:Group with operators
1256:Complemented lattice
1091:Algebraic structures
791:orthomodular lattice
712:if for all elements
708:A lattice is called
590:The diamond lattice
469:orthocomplementation
437:improve this article
426:to meet Knowledge's
328:improve this article
294:Orthocomplementation
216:distributive lattice
165:complemented lattice
142:orthomodular lattice
126:orthocomplementation
68:complemented lattice
18:Orthomodular lattice
1974:Alexandrov topology
1920:Lexicographic order
1879:Well-quasi-ordering
1367:Composition algebra
1127:Quasigroup and loop
550:inner product space
102: = 1 and
1955:Transitive closure
1915:Converse/Transpose
1624:Dilworth's theorem
1073:. Oliver and Boyd.
854:observed that the
183:, i.e. an element
90:, i.e. an element
56:
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2141:Partially ordered
1950:Symmetric closure
1935:Reflexive closure
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1061:978-0-12-295750-5
1042:978-0-7167-0442-3
1001:978-981-283-454-6
844:quantum mechanics
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428:quality standards
419:This article may
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300:De Morgan algebra
16:(Redirected from
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1925:Linear extension
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1279:Map of lattices
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1593:Weak ordering
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1578:Partial order
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966:, p. 11.
965:
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932:9780521461054
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868:
864:
860:
857:
856:propositional
853:
849:
845:
841:
838:
837:Hilbert space
834:
833:quantum logic
829:
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652:Hilbert space
650:
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638:quantum logic
635:
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345: –
344:
340:
339:Find sources:
333:
329:
323:
322:
317:This section
315:
311:
306:
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170:
169:least element
166:
158:
156:
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145:
143:
140:is called an
139:
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131:
127:
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120:
116:
111:
109:
106: ∧
105:
101:
98: ∨
97:
93:
89:
85:
81:
77:
76:least element
73:
70:is a bounded
69:
65:
61:
45:
33:
32:Hasse diagram
29:
19:
1967:& Orders
1945:Star product
1874:Well-founded
1827:Prefix order
1783:Distributive
1773:Complemented
1772:
1743:Foundational
1708:Completeness
1664:Zorn's lemma
1568:Cyclic order
1551:Key concepts
1521:Order theory
1480:
1466:
1452:
1438:
1387:Hopf algebra
1325:
1318:Vector space
1283:
1255:
1223:
1152:Group theory
1150:
1115: /
1070:
1051:
1032:
1020:
990:
983:
971:
964:Stern (1999)
959:
950:
941:
921:
914:
905:
878:
874:
870:
863:set products
830:
827:
820:
816:
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808:
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790:
786:
782:
780:
772:
766:
759:
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751:
747:
743:
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735:
731:
721:
717:
713:
707:
696:
692:
688:
684:
678:
674:
670:
666:
656:
640:, where the
632:
610:
609:The lattice
591:
572:
546:ortholattice
545:
541:
539:
533:
529:
525:
521:
511:
507:
498:
494:
490:
486:
476:
472:
468:
466:
451:
442:
435:Please help
431:
420:
390:
381:
371:
364:
357:
350:
338:
326:Please help
321:verification
318:
288:vector space
281:
276:
272:
270:
262:
258:
254:
250:
246:
242:
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229:
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146:
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125:
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114:
112:
107:
103:
99:
95:
91:
87:
83:
67:
64:order theory
60:mathematical
57:
2151:Riesz space
2112:Isomorphism
1988:Normal cone
1910:Composition
1844:Semilattice
1753:Homogeneous
1738:Equivalence
1588:Total order
1372:Lie algebra
1357:Associative
1261:Total order
1251:Semilattice
1225:Ring theory
867:linear sums
840:formulation
445:August 2014
439:if you can.
384:August 2014
147:In bounded
138:modular law
94:satisfying
38:and a line
2119:Order type
2053:Cofinality
1894:Well-order
1869:Transitive
1758:Idempotent
1691:Asymmetric
1482:PlanetMath
1468:PlanetMath
1454:PlanetMath
1440:PlanetMath
1013:References
552:, and the
354:newspapers
298:See also:
236:such that
187:such that
181:complement
130:involution
88:complement
44:Fano plane
2170:Upper set
2107:Embedding
2043:Antichain
1864:Tolerance
1854:Symmetric
1849:Semiorder
1795:Reflexive
1713:Connected
1382:Bialgebra
1188:Near-ring
1145:Lie group
1113:Semigroup
649:separable
645:subspaces
284:subspaces
2190:Category
1965:Topology
1832:Preorder
1815:Eulerian
1778:Complete
1728:Directed
1718:Covering
1583:Preorder
1542:Category
1537:Glossary
1218:Lie ring
1183:Semiring
1031:(1971).
885:See also
859:calculus
493:= 1 and
421:require
132:that is
119:interval
2070:Duality
2048:Cofinal
2036:Related
2015:Fréchet
1892:)
1768:Bounded
1763:Lattice
1736:)
1734:Partial
1602:Results
1573:Lattice
1349:Algebra
1341:Algebra
1246:Lattice
1237:Lattice
828:holds.
807:, then
738:, then
710:modular
423:cleanup
368:scholar
72:lattice
58:In the
42:of the
2095:Subnet
2075:Filter
2025:Normed
2010:Banach
1976:&
1883:Better
1820:Strict
1810:Graded
1701:topics
1532:Topics
1377:Graded
1308:Module
1299:Module
1198:Domain
1117:Monoid
1058:
1039:
998:
929:
642:closed
370:
363:
356:
349:
341:
179:has a
171:0 and
86:has a
78:0 and
74:(with
2085:Ideal
2063:Graph
1859:Total
1837:Total
1723:Dense
1343:-like
1301:-like
1239:-like
1208:Field
1166:-like
1140:Magma
1108:Group
1102:-like
1100:Group
897:Notes
789:. An
750:) = (
647:of a
528:then
375:JSTOR
361:books
286:of a
1676:list
1173:Ring
1164:Ring
1056:ISBN
1037:ISBN
996:ISBN
927:ISBN
877:and
850:and
819:) =
758:) ∧
720:and
691:) =
673:) =
501:= 0.
347:news
208:= 0.
66:, a
2090:Net
1890:Pre
1178:Rng
879:not
871:and
842:of
811:∨ (
799:if
742:∨ (
730:if
544:or
540:An
520:if
467:An
330:by
124:An
2192::
1479:.
1465:.
1451:.
1437:.
875:or
873:,
865:,
846:.
815:∧
803:≤
785:=
754:∨
746:∧
734:≤
716:,
695:∨
687:∧
677:∧
669:∨
661::
532:≤
524:≤
510:=
497:∧
489:∨
261:=
257:∧
249:=
245:∨
204:∧
196:∨
163:A
155:.
144:.
113:A
1888:(
1885:)
1881:(
1732:(
1679:)
1513:e
1506:t
1499:v
1485:.
1471:.
1457:.
1443:.
1417:e
1410:t
1403:v
1064:.
1045:.
1004:.
978:.
936:.
821:c
817:c
813:a
809:a
805:c
801:a
787:a
783:b
776:3
773:M
760:c
756:b
752:a
748:c
744:b
740:a
736:c
732:a
722:c
718:b
714:a
699:.
697:b
693:a
689:b
685:a
683:(
679:b
675:a
671:b
667:a
665:(
614:4
611:M
595:3
592:M
576:5
573:N
536:.
534:a
530:b
526:b
522:a
514:.
512:a
508:a
499:a
495:a
491:a
487:a
477:a
473:a
458:)
452:(
447:)
443:(
397:)
391:(
386:)
382:(
372:·
365:·
358:·
351:·
324:.
277:a
273:b
265:.
263:c
259:b
255:a
251:d
247:b
243:a
234:b
230:a
206:b
202:a
198:b
194:a
185:b
177:a
108:b
104:a
100:b
96:a
92:b
84:a
54:.
52:l
48:p
40:l
36:p
20:)
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