1933:
1760:
1706:
Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval will hold in every MV-algebra. Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras. Equivalently, Chang's completeness
1971:
by establishing a bijective correspondence between all isomorphism classes of approximately finite-dimensional C*-algebras with lattice-ordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include:
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242:
53:
1685:
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76:
903:
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217:
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96:
1968:
1581:
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601:
1805:
1604:
99:
745:
2409:
2215:
1247:. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of
2428:
Cabrer, L. M. & Mundici, D. A Stone-Weierstrass theorem for MV-algebras and unital â-groups. Journal of Logic and
Computation (2014).
2454:
1486:. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.
2418:
Mundici, D. The C*-Algebras of Three-Valued Logic. Logic
Colloquium '88, Proceedings of the Colloquium held in Padova 61â77 (1989).
1795:
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The way the MV-algebra characterizes all possible MV-algebras parallels the well-known fact that identities holding in the
807:
2483:
2394:
1741:
as an alternative model for the infinite-valued Ćukasiewicz logic. Wajsberg algebras and MV-algebras are term-equivalent.
142:
1584:
is an MV-algebra. Conversely, any MV-algebra is a lattice-ordered effect algebra with the Riesz decomposition property.
1727:
1715:
1083:
729:
721:
1270:
2099:
There are multiple frameworks implementing fuzzy logic (type II), and most of them implement what has been called a
2493:
2376:
Mundici, D.: Interpretation of AF C*-algebras in Ćukasiewicz sentential calculus. J. Funct. Anal. 65, 15â63 (1986)
1821:
1699:. If the standard MV-algebra over is employed, the set of all -tautologies determines so-called infinite-valued
1198:
681:
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558:
1570:
1828:-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is a faithful model only for the â”
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522:
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29:
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Di Nola A., Lettieri A. (1993) "Equational characterization of all varieties of MV-algebras,"
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hold in all possible
Boolean algebras. Moreover, MV-algebras characterize infinite-valued
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to the axioms defining an MV-algebra results in an axiomatization of
Boolean algebras.
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MV-algebra has the only element 0 and the operations defined in the only possible way,
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The Morita-equivalence between MV-algebras and abelian â-groups with strong unit
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997:
668:{\displaystyle \lnot (\lnot x\oplus y)\oplus y=\lnot (\lnot y\oplus x)\oplus x.}
121:
118:
21:
2256:
R. Cignoli, Proper n-Valued Ćukasiewicz
Algebras as S-Algebras of Ćukasiewicz
2141:
1421:
725:
2149:
1875:-algebras are MV-algebras that satisfy some additional axioms, just like the
2433:
191:
2363:
2330:------ (1959) "A new proof of the completeness of the Lukasiewicz axioms,"
1267:
equidistant real numbers between 0 and 1 (both included), that is, the set
2240:-valued ĆukasiewiczâMoisil algebrasâI. Discrete Math. 181, 155â177 (1998)
724:
of algebras. The variety of MV-algebras is a subvariety of the variety of
1573:
between lattice-ordered abelian groups with strong unit and MV-algebras.
797:{\displaystyle \langle L,\wedge ,\vee ,\otimes ,\rightarrow ,0,1\rangle }
1840:-valued Ćukasiewicz logics, suitable algebras were published in 1977 by
2265:
2172:"citing J. M. Font, A. J. Rodriguez, A. Torrens, "Wajsberg Algebras",
717:
2442:
1400:
of the standard MV-algebra; these algebras are usually denoted MV
2320:
Chang, C. C. (1958) "Algebraic analysis of many-valued logics,"
1879:-valued Ćukasiewicz logics have additional axioms added to the â”
2260:-Valued Propositional Calculi, Studia Logica, 41, 1982, 3-16,
2120:
Foulis, D. J. (2000-10-01). "MV and
Heyting Effect Algebras".
1927:
1754:
1243:
algebra corresponding to the three-valued Ćukasiewicz logic Ć
2103:. This is no more than the implementation of an MV-algebra.
1707:
theorem says that MV-algebras characterize infinite-valued
1489:
Daniele
Mundici extended the above construction to abelian
1900:-valued Ćukasiewicz logic; Cignoli called his discovery
433:{\displaystyle (x\oplus y)\oplus z=x\oplus (y\oplus z),}
1944:
1771:
1427:
Chang also constructed an MV-algebra from an arbitrary
1890:
published some additional constraints that added to LM
850:{\displaystyle x\vee y=(x\rightarrow y)\rightarrow y.}
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1836:. For the axiomatically more complicated (finitely)
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In 1984, Font, Rodriguez and
Torrens introduced the
1004:, as it forms the standard real-valued semantics of
179:{\displaystyle \langle A,\oplus ,\lnot ,0\rangle ,}
2050:
2002:
1816:. However, in 1956, Alan Rose discovered that for
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739:1998) as a prelinear commutative bounded integral
720:. Being defined by identities, MV-algebras form a
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2332:Transactions of the American Mathematical Society
2322:Transactions of the American Mathematical Society
117:. MV-algebras coincide with the class of bounded
98:, satisfying certain axioms. MV-algebras are the
1160:with Boolean negation. In fact adding the axiom
924:
1353:{\displaystyle \{0,1/(n-1),2/(n-1),\dots ,1\},}
2346:Algebraic Foundations of Many-valued Reasoning
2208:Non-commutative Multiple-Valued Logic Algebras
2402:Advanced Ćukasiewicz calculus and MV-algebras
1820:â„ 5, the ĆukasiewiczâMoisil algebra does not
1770: with: the extra axioms. You can help by
1603:in 1920. In particular, MV-algebras form the
8:
1981:approximately finite-dimensional C*-algebra
1969:approximately finite-dimensional C*-algebras
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1104:
1092:
791:
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703:
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146:
1232:{\displaystyle x\oplus x\oplus x=x\oplus x}
735:An MV-algebra can equivalently be defined (
709:{\displaystyle \langle A,\oplus ,0\rangle }
2041:
2040:
2031:
2025:
1996:
1995:
1993:
1916:-algebras are precisely Cignoli's proper
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1497:is such a group with strong (order) unit
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589:{\displaystyle x\oplus \lnot 0=\lnot 0,}
2112:
1862:-algebras; the inclusion is strict for
2307:: CS1 maint: archived copy as title (
2300:
2230:Iorgulescu, A.: Connections between MV
1722:in a manner analogous to the way that
1711:, defined as the set of -tautologies.
1424:ω) and their co-infinitesimals.
1360:which is closed under the operations
951:{\displaystyle x\oplus y=\min(x+y,1)}
678:By virtue of the first three axioms,
7:
1902:proper n-valued Ćukasiewicz algebras
1667:. Formulas mapped to 1 (that is, to
1580:that is lattice-ordered and has the
1454:}, which becomes an MV-algebra with
511:{\displaystyle x\oplus y=y\oplus x,}
2455:Stanford Encyclopedia of Philosophy
2051:{\displaystyle M_{n}(\mathbb {C} )}
1569:). This construction establishes a
804:satisfying the additional identity
1896:-algebras yield proper models for
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65:
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1832:-valued (infinitely-many-valued)
1804:-algebras) in the hope of giving
1931:
1758:
1000:, this MV-algebra is called the
547:{\displaystyle \lnot \lnot x=x,}
2210:. Springer. pp. viiâviii.
2095:Multi-adjoint logic programming
1924:Relation to functional analysis
1656:{\displaystyle \oplus ,\lnot ,}
1634:(in the language consisting of
1438:and defining the segment as {
1239:, then the axioms define the MV
2371:Metamathematics of Fuzzy Logic
2206:Lavinia Corina Ciungu (2013).
2045:
2037:
1920:-valued Ćukasiewicz algebras.
1856:-algebras are a subclass of LM
1329:
1317:
1303:
1291:
1195:If instead the axiom added is
945:
927:
889:
877:
864:A simple numerical example is
838:
835:
829:
823:
776:
653:
638:
623:
608:
424:
412:
394:
382:
368:which satisfies the following
106:; the letters MV refer to the
1:
2424:10.1016/s0049-237x(08)70262-3
2395:MV-ALGEBRAS. A short tutorial
2246:10.1016/S0012-365X(97)00052-6
1595:devised MV-algebras to study
1588:Relation to Ćukasiewicz logic
1501:, then the "unit interval" {
1434:by fixing a positive element
1429:totally ordered abelian group
1409:Another important example is
2382:10.1016/0022-1236(86)90015-7
2003:{\displaystyle \mathbb {C} }
1963:MV-algebras were related by
1582:Riesz decomposition property
989:{\displaystyle \lnot x=1-x.}
469:{\displaystyle x\oplus 0=x,}
1796:ĆukasiewiczâMoisil algebras
1716:two-element Boolean algebra
1185:{\displaystyle x\oplus x=x}
1084:two-element Boolean algebra
1082:MV-algebra is actually the
1040:{\displaystyle 0\oplus 0=0}
728:-algebras and contains all
2510:
2441:, Anna Carla Russo (2014)
2092:
1910:-algebras that are also MV
1069:{\displaystyle \lnot 0=0.}
1732:LindenbaumâTarski algebra
1834:ĆukasiewiczâTarski logic
1517:} can be equipped with ÂŹ
1113:{\displaystyle \{0,1\},}
2142:10.1023/A:1026454318245
1726:characterize classical
1691:-valuations are called
1571:categorical equivalence
1373:{\displaystyle \oplus }
1133:{\displaystyle \oplus }
860:Examples of MV-algebras
237:{\displaystyle \oplus }
48:{\displaystyle \oplus }
2364:10.1006/jabr.1999.7900
2176:, VIII, 1, 5-31, 1984"
2122:Foundations of Physics
2052:
2004:
1681:
1680:{\displaystyle \lnot }
1657:
1632:propositional formulas
1611:, as described below.
1491:lattice-ordered groups
1394:
1393:{\displaystyle \lnot }
1374:
1354:
1261:
1233:
1186:
1154:
1153:{\displaystyle \lnot }
1134:
1114:
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990:
952:
899:
851:
798:
710:
669:
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548:
512:
470:
434:
359:
332:
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286:
285:{\displaystyle \lnot }
261:
238:
213:
180:
92:
72:
49:
2434:10.1093/logcom/exu023
2344:, Mundici, D. (2000)
2342:D'Ottaviano, I. M. L.
2093:Further information:
2053:
2005:
1682:
1658:
1416:, consisting just of
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1234:
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991:
953:
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549:
513:
471:
435:
360:
333:
310:
287:
262:
239:
214:
181:
93:
73:
71:{\displaystyle \neg }
50:
2484:Algebraic structures
2024:
1992:
1978:Countable MV algebra
1671:
1638:
1630:from the algebra of
1614:Given an MV-algebra
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1364:
1271:
1251:
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2400:D. Mundici (2011).
2369:HĂĄjek, Petr (1998)
2340:Cignoli, R. L. O.,
2134:2000FoPh...30.1687F
2101:multi-adjoint logic
2074:finite-dimensional
1808:for the (finitely)
1806:algebraic semantics
1605:algebraic semantics
1002:standard MV-algebra
898:{\displaystyle A=,}
138:algebraic structure
100:algebraic semantics
78:, and the constant
30:algebraic structure
20:, a branch of pure
2463:Siegfried Gottwald
2353:Journal of Algebra
2266:10.1007/BF00373490
2048:
2000:
1943:. You can help by
1677:
1653:
1597:many-valued logics
1390:
1370:
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1182:
1150:
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741:residuated lattice
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665:
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358:{\displaystyle A,}
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308:{\displaystyle A,}
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260:{\displaystyle A,}
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212:{\displaystyle A,}
209:
176:
88:
68:
45:
2494:Many-valued logic
2459:Many-valued logic
2411:978-94-007-0839-6
2393:Daniele Mundici,
2217:978-3-319-01589-7
2128:(10): 1687â1706.
2086:
2085:
2066:complex matrices
1961:
1960:
1814:Ćukasiewicz logic
1788:
1787:
1720:Ćukasiewicz logic
1709:Ćukasiewicz logic
1701:Ćukasiewicz logic
1609:Ćukasiewicz logic
1260:{\displaystyle n}
1006:Ćukasiewicz logic
716:is a commutative
338:denoting a fixed
331:{\displaystyle 0}
104:Ćukasiewicz logic
91:{\displaystyle 0}
2501:
2439:Olivia Caramello
2415:
2313:
2312:
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2298:
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2289:
2283:. Archived from
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2222:
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2197:
2196:
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2181:. Archived from
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2009:
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1824:the Ćukasiewicz
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1739:Wajsberg algebra
1724:Boolean algebras
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1599:, introduced by
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2479:Algebraic logic
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2388:Further reading
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2278:"Archived copy"
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1990:
1989:
1965:Daniele Mundici
1957:
1951:
1948:
1941:needs expansion
1926:
1915:
1909:
1895:
1888:Roberto Cignoli
1883:-valued logic.
1882:
1874:
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1855:
1849:
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1803:
1794:introduced his
1784:
1778:
1775:
1768:needs expansion
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1601:Jan Ćukasiewicz
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2449:External links
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2236:-algebras and
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1792:Grigore Moisil
1790:In the 1940s,
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1746:
1743:
1728:bivalent logic
1676:
1652:
1649:
1646:
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1578:effect algebra
1554:
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1418:infinitesimals
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327:
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267:
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186:consisting of
175:
172:
169:
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163:
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154:
151:
148:
129:
126:
87:
67:
44:
13:
10:
9:
6:
4:
3:
2:
2506:
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2336:
2333:
2329:
2326:
2323:
2319:
2318:
2310:
2304:
2290:on 2014-08-10
2286:
2279:
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2250:
2247:
2243:
2239:
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2227:
2224:
2219:
2213:
2209:
2202:
2199:
2188:on 2014-08-10
2184:
2177:
2175:
2167:
2164:
2159:
2155:
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2147:
2143:
2139:
2135:
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2127:
2123:
2116:
2113:
2106:
2104:
2102:
2096:
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2081:
2078:
2077:
2073:
2070:
2069:
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2032:
2028:
2020:
2017:
2013:
2012:
1988:
1985:
1984:
1980:
1977:
1976:
1973:
1970:
1966:
1955:
1952:November 2012
1946:
1942:
1939:This section
1937:
1934:
1930:
1929:
1923:
1921:
1919:
1914:
1908:
1903:
1899:
1894:
1889:
1884:
1878:
1873:
1867:
1865:
1860:
1854:
1850:-algebras. MV
1848:
1844:and called MV
1843:
1839:
1835:
1827:
1823:
1819:
1815:
1811:
1807:
1802:
1797:
1793:
1782:
1773:
1769:
1766:This section
1764:
1761:
1757:
1756:
1749:
1744:
1742:
1740:
1735:
1733:
1729:
1725:
1721:
1717:
1712:
1710:
1704:
1702:
1698:
1694:
1690:
1666:
1650:
1644:
1641:
1633:
1629:
1625:
1621:
1617:
1612:
1610:
1606:
1602:
1598:
1594:
1587:
1585:
1583:
1579:
1574:
1572:
1568:
1564:
1560:
1552:
1548:
1545:(x + y), and
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1487:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1430:
1425:
1423:
1419:
1415:
1414:'s MV-algebra
1413:
1407:
1404:
1367:
1347:
1341:
1338:
1335:
1332:
1326:
1323:
1320:
1313:
1309:
1306:
1300:
1297:
1294:
1287:
1283:
1280:
1277:
1254:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1193:
1179:
1176:
1173:
1170:
1167:
1127:
1107:
1101:
1098:
1095:
1085:
1081:
1076:
1063:
1060:
1057:
1034:
1031:
1028:
1025:
1022:
1014:
1009:
1007:
1003:
999:
983:
980:
977:
974:
971:
968:
942:
939:
936:
933:
930:
921:
918:
915:
912:
892:
886:
883:
880:
874:
871:
859:
857:
844:
841:
832:
826:
820:
817:
814:
811:
788:
785:
782:
779:
773:
770:
767:
764:
761:
758:
755:
752:
742:
738:
733:
731:
727:
723:
719:
700:
697:
694:
691:
688:
662:
659:
656:
650:
647:
644:
632:
629:
626:
620:
617:
614:
598:
583:
580:
574:
571:
565:
562:
555:
541:
538:
535:
532:
519:
505:
502:
499:
496:
493:
490:
487:
484:
477:
463:
460:
457:
454:
451:
448:
441:
427:
421:
418:
415:
409:
406:
403:
400:
397:
391:
388:
385:
375:
374:
373:
371:
352:
349:
341:
325:
317:
302:
299:
272:
268:
254:
251:
231:
224:
220:
206:
203:
196:
193:
189:
188:
187:
173:
167:
164:
158:
155:
152:
149:
139:
135:
127:
125:
123:
120:
116:
112:
110:
105:
101:
85:
58:
42:
35:
31:
27:
23:
19:
2404:. Springer.
2401:
2370:
2355:
2352:
2345:
2334:
2331:
2324:
2321:
2292:. Retrieved
2285:the original
2272:
2257:
2252:
2237:
2232:
2226:
2207:
2201:
2190:. Retrieved
2183:the original
2173:
2166:
2125:
2121:
2115:
2100:
2098:
2082:commutative
2063:
2059:
2015:
1962:
1949:
1945:adding to it
1940:
1917:
1912:
1906:
1901:
1897:
1892:
1885:
1876:
1871:
1868:
1863:
1858:
1852:
1846:
1837:
1825:
1817:
1809:
1800:
1789:
1776:
1772:adding to it
1767:
1747:
1736:
1713:
1705:
1692:
1688:
1664:
1663:and 0) into
1628:homomorphism
1619:
1615:
1613:
1591:
1575:
1566:
1562:
1558:
1550:
1546:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1488:
1483:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1426:
1410:
1408:
1402:
1194:
1079:
1077:
1012:
1010:
1001:
863:
734:
677:
367:
133:
131:
122:BCK algebras
108:
25:
15:
2489:Fuzzy logic
2174:Stochastica
2089:In software
1779:August 2014
1697:tautologies
1687:0) for all
1593:C. C. Chang
1080:two-element
998:fuzzy logic
318:a constant
128:Definitions
119:commutative
115:Ćukasiewicz
109:many-valued
22:mathematics
2473:Categories
2358:: 463â474
2327:: 476â490.
2294:2014-08-21
2192:2014-08-21
2107:References
2018:, ..., 1 }
1422:order type
1420:(with the
370:identities
134:MV-algebra
26:MV-algebra
2373:. Kluwer.
2348:. Kluwer.
2158:116763476
2150:1572-9516
1886:In 1982,
1751:-algebras
1675:¬
1648:¬
1642:⊕
1624:valuation
1388:¬
1368:⊕
1336:…
1324:−
1298:−
1224:⊕
1212:⊕
1206:⊕
1171:⊕
1148:¬
1128:⊕
1055:¬
1026:⊕
978:−
966:¬
916:⊕
839:→
830:→
815:∨
792:⟩
777:→
771:⊗
765:∨
759:∧
750:⟨
704:⟩
695:⊕
686:⟨
657:⊕
648:⊕
642:¬
636:¬
627:⊕
618:⊕
612:¬
606:¬
578:¬
569:¬
566:⊕
530:¬
527:¬
500:⊕
488:⊕
452:⊕
419:⊕
410:⊕
398:⊕
389:⊕
280:¬
232:⊕
192:non-empty
171:⟩
162:¬
156:⊕
147:⟨
66:¬
43:⊕
2337:: 74â80.
2303:cite web
1904:. The LM
1812:-valued
2130:Bibcode
2079:boolean
2058:, i.e.
1474:) and ÂŹ
1013:trivial
722:variety
340:element
32:with a
2408:
2214:
2156:
2148:
2071:finite
2014:{0, 1/
1986:{0, 1}
1869:The MV
1509:| 0 â€
1462:= min(
1446:| 0 â€
718:monoid
136:is an
28:is an
2461:"âby
2288:(PDF)
2281:(PDF)
2186:(PDF)
2179:(PDF)
2154:S2CID
1866:â„ 5.
1822:model
1730:(see
1626:is a
1618:, an
1553:= 0 âš
1493:. If
1412:Chang
1120:with
737:HĂĄjek
111:logic
24:, an
2406:ISBN
2309:link
2212:ISBN
2146:ISSN
1380:and
1078:The
1047:and
1011:The
958:and
55:, a
2457:: "
2430:doi
2420:doi
2378:doi
2360:doi
2356:221
2262:doi
2242:doi
2138:doi
1967:to
1947:.
1798:(LM
1774:.
1734:).
1607:of
1576:An
925:min
596:and
342:of
315:and
292:on
244:on
195:set
132:An
113:of
102:of
16:In
2475::
2335:88
2325:88
2305:}}
2301:{{
2152:.
2144:.
2136:.
2126:30
2124:.
1745:MV
1703:.
1565:â
1561:+
1549:â
1537:=
1533:â
1529:,
1525:â
1521:=
1513:â€
1505:â
1482:â
1478:=
1470:+
1466:,
1458:â
1450:â€
1442:â
1406:.
1064:0.
1008:.
732:.
726:BL
372::
269:a
221:a
190:a
124:.
2465:.
2432::
2422::
2414:.
2380::
2366:.
2362::
2311:)
2297:.
2264::
2258:n
2244::
2238:n
2233:n
2220:.
2195:.
2160:.
2140::
2132::
2064:n
2062:Ă
2060:n
2046:)
2042:C
2038:(
2033:n
2029:M
2016:n
1997:C
1954:)
1950:(
1918:n
1913:n
1907:n
1898:n
1893:n
1881:0
1877:n
1872:n
1864:n
1859:n
1853:n
1847:n
1838:n
1830:0
1826:n
1818:n
1810:n
1801:n
1781:)
1777:(
1748:n
1695:-
1693:A
1689:A
1665:A
1651:,
1645:,
1622:-
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1177:=
1174:x
1168:x
1108:,
1105:}
1102:1
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1058:0
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943:1
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884:,
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878:[
875:=
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