142:
If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as
131:
Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a dual
951:
929:
907:
880:
763:
618:
542:
508:
439:. If the pseudocomplement satisfies the law of the excluded middle, the resulting algebra is also Boolean. However, if only the weaker law ¬
223:-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of
834:
564:
423:= 0 (i.e. the law of noncontradiction) but to drop the law of the excluded middle and the law of double negation. This approach (called
124:
do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a
920:; Walker, C.; Walker, E. (2003). "Fuzzy Logics Arising From Strict De Morgan Systems". In Rodabaugh, S. E.; Klement, E. P. (eds.).
673:
224:
533:
Béziau, Jean-Yves (2012). "A History of Truth-Values". In Gabbay, Dov M.; Pelletier, Francis Jeffry; Woods, John (eds.).
922:
Topological and
Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets
381:
125:
890:
Cattaneo, G.; Ciucci, D. (2009). "Lattices with
Interior and Closure Operators and Abstract Approximation Spaces".
102:
200:
464:
985:
980:
970:
939:
78:
875:. Springer Science & Business Media. Part II. Chapter 6. Basic Logico-Algebraic Structures, pp. 193-210.
975:
117:
258:) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold.
415:
De Morgan algebras are not the only plausible way to generalize
Boolean algebras. Another way is to keep ¬
377:
384:
also meet this definition of Kleene algebra. The simplest Kleene algebra that is not
Boolean is Kleene's
369:
216:
195:
around 1935, although without the restriction of having a 0 and a 1. They were then variously called
155:
47:
754:
Balbes, Raymond; Dwinger, Philip (1975). "Chapter IX. De Morgan
Algebras and Lukasiewicz Algebras".
634:
397:
82:
373:
873:
A Geometry of
Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns
859:
800:
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385:
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28:
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947:
925:
903:
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849:
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436:
432:
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403:
262:
152:
44:
493:
350:
204:
192:
854:
584:
964:
825:
448:
132:
812:
Batyrshin, I.Z. (1990). "On fuzzinesstic measures of entropy on Kleene algebras".
899:
428:
231:
20:
917:
781:
Annales scientifiques de l'Université de Jassy, vol. 22 (1936), pp. 1–118
230:
De Morgan algebras are important for the study of the mathematical aspects of
894:. Lecture Notes in Computer Science 67–116. Vol. 5656. pp. 67–116.
735:
718:
710:
693:
451:. More generally, both De Morgan and Stone algebras are proper subclasses of
863:
804:
658:
593:
407:(1938). The algebra was named after Kleene by Brignole and Monteiro.
265:'s four-valued semantics for De Morgan algebra, which has the values
796:
650:
676:(1964). "Caracterisation des algèbres de Nelson par des egalités".
723:
Proceedings of the Japan
Academy, Series A, Mathematical Sciences
698:
Proceedings of the Japan
Academy, Series A, Mathematical Sciences
686:
A (possibly abbreviated) version of this paper appeared later in
368:
Examples of Kleene algebras in the sense defined above include:
35:= (A, ∨, ∧, 0, 1, ¬) such that:
944:
Reasoning in
Quantum Theory: Sharp and Unsharp Quantum Logics
719:"Caracterisation des algèbres de Nelson par des egalités, II"
694:"Caracterisation des algèbres de Nelson par des egalités, I"
537:. North Holland (an imprint of Elsevier). pp. 280–281.
684:. Instituto de Matematica Universidad del sur Bahia Blanca.
31:, a British mathematician and logician) is a structure
143:
an involutive anti-automorphism, that is, a structure
349:. (This notion should not be confused with the other
842:Transactions of the American Mathematical Society
607:Kaarli, Kalle; Pixley, Alden F. (21 July 2000).
572:Proceedings of the American Mathematical Society
942:; Giuntini, Roberto; Greechie, Richard (2004).
16:System of logic lacking the excluded middle law
329:If a De Morgan algebra additionally satisfies
8:
871:Pagliani, Piero; Chakraborty, Mihir (2008).
610:Polynomial Completeness in Algebraic Systems
254:), 0, 1, 1 −
717:Brignole, Diana; Monteiro, Antonio (1967).
692:Brignole, Diana; Monteiro, Antonio (1967).
637:(1938). "On Notation for Ordinal Numbers".
558:
556:
554:
853:
734:
709:
583:
565:"Injective de Morgan and Kleene Algebras"
535:Logic: A History of its Central Concepts
778:Recherches sur l'algèbre de la logique.
476:
43:, ∨, ∧, 0, 1) is a
431:; if the set of semicomplements has a
357:.) This notion has also been called a
191:De Morgan algebras were introduced by
147:= (A, ≤, ¬) such that:
528:
526:
524:
522:
520:
7:
499:. Oxford University Press. pp.
491:Blyth, T. S.; Varlet, J. C. (1994).
486:
484:
482:
480:
427:) is well-defined even for a (meet)
139:, ∨, ∧, 0, 1).
14:
855:10.1090/S0002-9947-1958-0095135-X
776:(1936). "Reviews: Moisil Gr. C..
585:10.1090/S0002-9939-1975-0357259-4
447:= 1 is required, this results in
89:In a De Morgan algebra, the laws
758:. University of Missouri Press.
688:Proceedings of the Japan Academy
53:¬ is a De Morgan involution: ¬(
1:
785:The Journal of Symbolic Logic
639:The Journal of Symbolic Logic
396:made its first appearance in
234:. The standard fuzzy algebra
892:Transactions on Rough Sets X
826:10.1016/0165-0114(90)90126-Q
613:. CRC Press. pp. 297–.
81:that additionally satisfies
900:10.1007/978-3-642-03281-3_3
1002:
835:"Lattices with involution"
678:Notas de Logica Matematica
103:law of the excluded middle
940:Dalla Chiara, Maria Luisa
563:Cignoli, Roberto (1975).
465:orthocomplemented lattice
118:law of noncontradiction
833:Kalman, J. A. (1958).
814:Fuzzy Sets and Systems
736:10.3792/pja/1195521625
711:10.3792/pja/1195521624
370:lattice-ordered groups
197:quasi-boolean algebras
756:Distributive lattices
435:it is usually called
378:Łukasiewicz algebras
156:distributive lattice
48:distributive lattice
425:semicomplementation
355:regular expressions
320:are not comparable.
261:Another example is
238:= (, max(
386:three-valued logic
29:Augustus De Morgan
953:978-1-4020-1978-4
931:978-1-4020-1515-1
909:978-3-642-03280-6
882:978-1-4020-8622-9
765:978-0-8262-0163-8
620:978-1-58488-203-9
544:978-0-08-093170-8
510:978-0-19-859938-8
345:, it is called a
151:(A, ≤) is a
25:De Morgan algebra
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437:pseudocomplement
433:greatest element
402:On notation for
382:Boolean algebras
281:(either), where
225:Antonio Monteiro
83:De Morgan's laws
1001:
1000:
996:
995:
994:
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991:
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986:Ockham algebras
981:Algebraic logic
971:Boolean algebra
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748:Further reading
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495:Ockham algebras
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453:Ockham algebras
443: ∨ ¬¬
413:
411:Related notions
404:ordinal numbers
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391:
327:
126:Boolean algebra
17:
12:
11:
5:
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997:
989:
988:
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976:Lattice theory
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848:(2): 485–491.
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672:Brignole, D.;
664:
645:(4): 150–155.
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578:(2): 269–278.
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475:
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472:
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467:
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449:Stone algebras
412:
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393:
389:
351:Kleene algebra
347:Kleene algebra
326:
325:Kleene algebra
323:
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321:
311:
297:
193:Grigore Moisil
189:
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87:
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65: ∨ ¬
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635:Kleene, S. C.
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419: ∧
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374:Post algebras
371:
366:
364:
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353:generalizing
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97: ∨
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84:
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57: ∧
56:
52:
49:
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42:
38:
37:
36:
34:
30:
27:(named after
26:
22:
946:. Springer.
943:
924:. Springer.
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891:
872:
845:
841:
820:(1): 47–60.
817:
813:
788:
784:
780:
777:
774:Birkhoff, G.
755:
726:
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681:
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674:Monteiro, A.
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246:), min(
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217:J. A. Kalman
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172:
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144:
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136:
133:automorphism
130:
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113:
109:
98:
94:
88:
77:. (i.e. an
74:
70:
66:
62:
58:
54:
40:
32:
24:
18:
429:semilattice
365:by Kalman.
277:(oth), and
232:fuzzy logic
21:mathematics
965:Categories
918:Gehrke, M.
471:References
203:, e.g. by
79:involution
791:(2): 63.
213:-lattices
207:and also
459:See also
363:-lattice
273:(alse),
864:1993112
805:2268551
659:2267778
594:2039730
359:normal
269:(rue),
250:,
242:,
205:Rasiowa
199:in the
153:bounded
45:bounded
950:
928:
906:
879:
862:
803:
762:
657:
617:
592:
541:
507:
398:Kleene
105:), and
69:and ¬¬
860:JSTOR
838:(PDF)
801:JSTOR
729:(4).
704:(4).
655:JSTOR
590:JSTOR
568:(PDF)
310:, and
306:<
302:<
292:<
288:<
169:, and
158:, and
116:= 0 (
101:= 1 (
61:) = ¬
50:, and
948:ISBN
926:ISBN
904:ISBN
877:ISBN
760:ISBN
615:ISBN
539:ISBN
505:ISBN
503:–5.
376:and
316:and
263:Dunn
135:of (
23:, a
896:doi
850:doi
822:doi
793:doi
783:".
731:doi
706:doi
647:doi
580:doi
400:'s
392:. K
341:∨ ¬
333:∧ ¬
219:. (
215:by
183:≤ ¬
179:→ ¬
19:In
967::
902:.
858:.
846:87
844:.
840:.
818:34
816:.
799:.
787:.
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725:.
721:.
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700:.
696:.
690::
682:20
680:.
653:.
641:.
588:.
576:47
574:.
570:.
553:^
519:^
479:^
455:.
380:.
372:,
337:≤
227:.
175:≤
165:=
161:¬¬
128:.
73:=
956:.
934:.
912:.
898::
885:.
866:.
852::
828:.
824::
807:.
795::
789:1
768:.
739:.
733::
714:,
708::
661:.
649::
643:3
623:.
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445:x
441:x
421:x
417:x
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390:3
388:K
361:i
343:y
339:y
335:x
331:x
318:N
314:B
308:T
304:N
300:F
296:,
294:T
290:B
286:F
279:N
275:B
271:F
267:T
256:x
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248:x
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236:F
221:i
211:i
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185:x
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120:)
114:x
110:x
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