1709:
1186:
1704:{\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}}
3423:
38:
877:
638:
1025:
722:
2941:, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.
1801:
with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called
2276:
2508:
2415:
434:
2326:
1086:
1181:
525:
659:
2134:
354:
2181:
460:
2919:
in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state.
896:
2649:. Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is,
2826:
IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in
2059:
1191:
2858:
650:. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an
872:{\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}}
3443:
500:
2944:
Another involution used in computers is an order-2 bitwise permutation. For example. a colour value stored as integers in the form
2682:
1030:
3073:
2681:
The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the
1988:, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise
2188:
3380:
3109:
3083:
1814:
31:
2426:
2333:
3448:
3130:
2884:. Since the converse of the converse is the original relation, the conversion operation is an involution on the
359:
3059:
2281:
504:
17:
2541:, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the
3407:
3372:
3162:
2897:
1798:
1741:
1138:
2833:
The involutiveness of negation is an important characterization property for logics and the corresponding
3427:
3402:
2838:
2834:
2777:
2675:
3212:
727:
2538:
1794:
1714:
655:
2104:
1774:
meet any line (not through a vertex) in three pairs of an involution. This theorem has been called
1730:
313:
196:
1104:
3208:
719:, then its graph is its own reflection. Some basic examples of involutions include the functions
212:
3397:
2806:
Generally in non-classical logics, negation that satisfies the law of double negation is called
1849:. Except for in characteristic 2, such operators are diagonalizable for a given basis with just
3248:
2893:
2142:
633:{\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.}
200:
162:
76:
3099:
2885:
2870:
2701:
439:
304:
231:
3268:
3191:
3140:
2850:
2610:
2587:
2101:, an (anti-)involution is defined by the following axioms: if we consider a transformation
2004:
1976:
1787:
647:
263:
238:
115:
3390:
3316:
Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions".
2862:
2819:
879:
These may be composed in various ways to produce additional involutions. For example, if
8:
3356:
2827:
2810:. In algebraic semantics, such a negation is realized as an involution on the algebra of
2606:
2579:
2518:
2074:
1993:
1989:
1771:
1734:
680:
676:
300:
3364:
3343:
3325:
3179:
2937:
This predates binary computers; practically all mechanical cipher machines implement a
2815:
2542:
2098:
2014:
3376:
3105:
3079:
3055:
2938:
2912:
2881:
2655:
1997:
1826:
1748:
1020:{\displaystyle f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}}
220:
176:
144:
3371:, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI:
3347:
3386:
3335:
3260:
3216:
3171:
2916:
2873:, and other pairs of important varieties of algebras (resp. corresponding logics).
2765:
2646:
2572:
2514:
2078:
204:
148:
83:
2915:
with a given value for one parameter is an involution on the other parameter. XOR
2754:
of involutions subject only to relations involving powers of pairs of elements of
2674:
is an involution if and only if it can be written as a finite product of disjoint
147:
is a trivial example of an involution. Examples of nontrivial involutions include
3280:
3227:
3187:
3136:
2877:
2846:
2842:
2786:
1857:
s on the diagonal of the corresponding matrix. If the operator is orthogonal (an
1737:. Performing a reflection twice brings a point back to its original coordinates.
1726:
228:
3339:
1778:'s Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to the
2761:
2086:
1930:, and that is the identity on all other basis vectors. It can be checked that
3437:
2927:
2889:
2745:
2583:
508:
2760:. Coxeter groups can be used, among other things, to describe the possible
292:
The number of involutions, including the identity involution, on a set with
3125:
3104:, Springer Science & Business Media, Problem 1.11(a), p. 27,
3021:
2602:
2575:
that is its own inverse function. Examples of involutions in common rings:
2028:
1760:
490:
3294:
2814:. Examples of logics that have involutive negation are Kleene and Bochvar
3360:
3026:
2923:
2823:
2811:
2671:
2564:
643:
486:
482:
478:
60:
1763:
of period 2, that is, a projectivity that interchanges pairs of points.
3183:
3153:
2866:
2854:
2082:
651:
474:
470:
466:
216:
192:
1127:
Other nonlinear examples can be constructed by wrapping an involution
3330:
2931:
2558:
2546:
2067:
1985:
1775:
137:
3175:
642:
The number of fixed points of an involution on a finite set and its
2781:
1744:; not a reflection in the above sense, and so, a distinct example.
208:
1975:
For a specific basis, any linear operator can be represented by a
665:
1809:
Another type of involution occurring in projective geometry is a
522:
can also be expressed by non-recursive formulas, such as the sum
55:
that, when applied twice, brings one back to the starting point.
3422:
3016:
2922:
Two special cases of this, which are also involutions, are the
1767:
Any projectivity that interchanges two points is an involution.
3251:(Cambridge and London: Harvard and Heinemann), pp. 610–3
3078:(2nd ed.), W. W. Norton & Company, Inc, p. 426,
3031:
1797:, it has another, and consists of the correspondence between
224:
37:
2420:
An anti-involution does not obey the last axiom but instead
1725:
A simple example of an involution of the three-dimensional
495:
2909:
2092:
1888:
are basis elements. There exists a linear transformation
3245:
3354:
1831:
In linear algebra, an involution is a linear operator
2429:
2336:
2284:
2191:
2145:
2107:
1864:
For example, suppose that a basis for a vector space
1189:
1141:
1033:
899:
725:
528:
442:
362:
316:
3135:, Reading, Mass.: Addison-Wesley, pp. 48, 65,
2926:operation which is XOR with an all-ones value, and
2837:. For instance, involutive negation characterizes
2502:
2409:
2320:
2270:
2175:
2128:
1703:
1175:
1080:
1027:is an involution, and more generally the function
1019:
871:
693:. This is due to the fact that the inverse of any
632:
454:
428:
348:
2663:was defined more broadly, and accordingly so was
3435:
2271:{\displaystyle f(x_{1}+x_{2})=f(x_{1})+f(x_{2})}
2003:The definition of involution extends readily to
3156:(1990), "A one-sentence proof that every prime
2849:arises by adding the law of double negation to
2537:. Taken as an axiom, it leads to the notion of
666:Involution throughout the fields of mathematics
2900:involution, it is preserved under conversion.
2503:{\displaystyle f(x_{1}x_{2})=f(x_{2})f(x_{1})}
2410:{\displaystyle f(x_{1}x_{2})=f(x_{1})f(x_{2})}
697:function will be its reflection over the line
32:Involution (disambiguation) § Mathematics
3318:Computers & Mathematics with Applications
287:
571:
558:
3315:
2896:. While this ordering is reversed with the
2853:. The same relationship holds also between
1713:Other elementary involutions are useful in
2930:encryption, which is an XOR with a secret
679:of an involution (on the real numbers) is
429:{\displaystyle a_{n}=a_{n-1}+(n-1)a_{n-2}}
3329:
2321:{\displaystyle f(\lambda x)=\lambda f(x)}
511:with a given number of cells. The number
465:The first few terms of this sequence are
3232:The Geometrical Work of Girard Desargues
3097:
2613:2; that is, an involution is an element
1103:. (This is the self-inverse subset of
1081:{\displaystyle g(x)={\frac {x+b}{cx-1}}}
715:. If, in particular, the function is an
670:
36:
3071:
1861:), it is orthonormally diagonalizable.
1770:The three pairs of opposite sides of a
1176:{\displaystyle f:=h^{-1}\circ g\circ h}
14:
3436:
3292:
3160:≡ 1 (mod 4) is a sum of two squares",
3152:
2683:classification of finite simple groups
2592:taking the transpose in a matrix ring.
2093:Quaternion algebra, groups, semigroups
2060:Involutions are related to idempotents
1754:
1747:These transformations are examples of
3203:
3201:
3124:
2771:
1720:
262:is an involution if and only if they
131:
2766:generalizations to higher dimensions
2513:This former law is sometimes called
507:, and they also count the number of
2903:
128:twice produces the original value.
24:
3309:
3198:
2659:. By the end of the 19th century,
1992:is an independent involution, the
25:
3460:
3415:
3133:, Volume 3: Sorting and Searching
2784:in classical logic satisfies the
1820:
707:. This can be seen by "swapping"
3444:Algebraic properties of elements
3421:
2861:(and so correspondingly between
2571:is customarily taken to mean an
2050:is the identity homomorphism on
503:); these numbers are called the
27:Function that is its own inverse
3286:
3273:
3247:, Volume II, number 362 in the
3131:The Art of Computer Programming
3101:The Elements of Operator Theory
2780:is an involution. Accordingly,
2776:The operation of complement in
2596:
1782:of Euclid in Volume VII of the
1664:
1624:
1088:is an involution for constants
3295:"The Mechanization of Ciphers"
3254:
3237:
3221:
3209:Elementary Projective Geometry
3146:
3118:
3091:
3065:
3044:
2748:are groups generated by a set
2552:
2497:
2484:
2478:
2465:
2456:
2433:
2404:
2391:
2385:
2372:
2363:
2340:
2315:
2309:
2297:
2288:
2265:
2252:
2243:
2230:
2221:
2195:
2164:
2161:
2155:
2149:
2123:
2117:
2111:
1674:
1668:
1634:
1628:
1587:
1581:
1552:
1546:
1521:
1515:
1468:
1462:
1432:
1426:
1385:
1379:
1310:
1304:
1274:
1268:
1244:
1238:
1203:
1197:
1043:
1037:
998:
992:
989:
963:
957:
951:
948:
922:
916:
910:
831:
825:
788:
782:
746:
740:
618:
603:
407:
395:
13:
1:
3373:American Mathematical Society
3234:, (New York: Springer), p. 54
3163:American Mathematical Monthly
3037:
2845:. Correspondingly, classical
2820:Łukasiewicz many-valued logic
2129:{\displaystyle x\mapsto f(x)}
1837:on a vector space, such that
1742:reflection through the origin
654:of elements has at least one
349:{\displaystyle a_{0}=a_{1}=1}
3243:Ivor Thomas (editor) (1980)
3098:Kubrusly, Carlos S. (2011),
2586:, and its equivalent in the
2136:then it is an involution if
1715:solving functional equations
660:Fermat's two squares theorem
658:. This can be used to prove
41:An involution is a function
7:
3403:Encyclopedia of Mathematics
3340:10.1016/j.camwa.2006.10.029
3010:
2609:is an involution if it has
2042:is called an involution if
1135:and its inverse, producing
10:
3465:
3281:"A Course on Group Theory"
3213:Cambridge University Press
3072:Russell, Bertrand (1903),
2704:if there is an involution
2556:
1824:
288:Involutions on finite sets
29:
3075:Principles of mathematics
3052:Calculus: Single Variable
2539:semigroup with involution
2176:{\displaystyle f(f(x))=x}
1793:If an involution has one
1131:in an arbitrary function
124:. Equivalently, applying
3265:Introduction to Geometry
3050:Robert Alexander Adams,
2972:, resulting in the form
2070:in a one-to-one manner.
2066:is invertible then they
3369:The book of involutions
2888:. Binary relations are
2880:, every relation has a
2183:(it is its own inverse)
2000:is also an involution.
455:{\displaystyle n>1.}
299:elements is given by a
227:transformation and the
3449:Functions and mappings
3283:. p. 10, section 1.13.
3249:Loeb Classical Library
3230:and J. J. Gray (1987)
2787:law of double negation
2504:
2411:
2322:
2272:
2177:
2130:
1740:Another involution is
1705:
1177:
1105:Möbius transformations
1082:
1021:
873:
634:
575:
456:
430:
350:
56:
3293:Goebel, Greg (2018).
3269:John Wiley & Sons
3207:A.G. Pickford (1909)
2886:category of relations
2835:varieties of algebras
2588:split-complex numbers
2557:Further information:
2517:. It also appears in
2505:
2412:
2323:
2273:
2178:
2131:
2085:are special types of
1984:. Every matrix has a
1859:orthogonal involution
1825:Further information:
1706:
1178:
1117:, then normalized to
1083:
1022:
874:
671:Real-valued functions
635:
542:
457:
431:
351:
305:Heinrich August Rothe
232:polyalphabetic cipher
73:self-inverse function
40:
3430:at Wikimedia Commons
3357:Merkurjev, Alexander
3299:Classical Cryptology
2851:intuitionistic logic
2427:
2334:
2282:
2189:
2143:
2105:
1966:is an involution of
1870:is chosen, and that
1788:Pappus of Alexandria
1187:
1139:
1031:
897:
723:
526:
440:
360:
314:
136:Any involution is a
30:For other uses, see
3365:Tignol, Jean-Pierre
2828:t-norm fuzzy logics
2816:three-valued logics
2580:complex conjugation
2549:as the involution.
2075:functional analysis
1994:conjugate transpose
1990:complex conjugation
1799:harmonic conjugates
1772:complete quadrangle
1759:An involution is a
1755:Projective geometry
301:recurrence relation
250:of two involutions
177:complex conjugation
69:involutory function
3355:Knus, Max-Albert;
2772:Mathematical logic
2653:was taken to mean
2605:, an element of a
2543:full linear monoid
2500:
2407:
2318:
2268:
2173:
2126:
2099:quaternion algebra
2089:with involutions.
1749:affine involutions
1721:Euclidean geometry
1701:
1699:
1173:
1078:
1017:
869:
867:
644:number of elements
630:
452:
426:
346:
221:reciprocal ciphers
132:General properties
57:
3426:Media related to
2960:, could exchange
2939:reciprocal cipher
2913:bitwise operation
2882:converse relation
2863:Łukasiewicz logic
2797:is equivalent to
2762:regular polyhedra
2656:permutation group
2079:Banach *-algebras
2007:. Given a module
1998:Hermitian adjoint
1827:Involutory matrix
1660:
1620:
1619:
1539:
1504:
1419:
1368:
1231:
1076:
1015:
860:
806:
625:
569:
505:telephone numbers
16:(Redirected from
3456:
3425:
3411:
3393:
3351:
3333:
3303:
3302:
3290:
3284:
3277:
3271:
3261:H. S. M. Coxeter
3258:
3252:
3241:
3235:
3225:
3219:
3217:Internet Archive
3205:
3196:
3194:
3150:
3144:
3143:
3126:Knuth, Donald E.
3122:
3116:
3114:
3095:
3089:
3088:
3069:
3063:
3048:
3006:
2987:
2971:
2965:
2959:
2904:Computer science
2878:binary relations
2876:In the study of
2865:and fuzzy logic
2843:Heyting algebras
2839:Boolean algebras
2802:
2796:
2778:Boolean algebras
2759:
2753:
2741:
2719:
2709:
2699:
2693:
2647:identity element
2644:
2638:
2628:
2618:
2573:antihomomorphism
2536:
2515:antidistributive
2509:
2507:
2506:
2501:
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2127:
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2012:
1983:
1971:
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1911:
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1123:
1116:
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1061:
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1018:
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876:
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859:
842:
824:
823:
807:
799:
781:
780:
739:
738:
714:
710:
706:
692:
683:across the line
639:
637:
636:
631:
626:
624:
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595:
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372:
371:
355:
353:
352:
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339:
338:
326:
325:
298:
283:
261:
255:
249:
205:circle inversion
190:
189:
174:
160:
127:
123:
113:
106:
82:that is its own
81:
54:
21:
3464:
3463:
3459:
3458:
3457:
3455:
3454:
3453:
3434:
3433:
3418:
3396:
3383:
3312:
3310:Further reading
3307:
3306:
3291:
3287:
3278:
3274:
3259:
3255:
3242:
3238:
3226:
3222:
3206:
3199:
3176:10.2307/2323918
3151:
3147:
3123:
3119:
3112:
3096:
3092:
3086:
3070:
3066:
3049:
3045:
3040:
3013:
2989:
2973:
2967:
2961:
2945:
2906:
2898:complementation
2798:
2791:
2774:
2755:
2749:
2721:
2711:
2705:
2695:
2689:
2640:
2630:
2620:
2614:
2599:
2561:
2555:
2522:
2491:
2487:
2472:
2468:
2450:
2446:
2440:
2436:
2428:
2425:
2424:
2398:
2394:
2379:
2375:
2357:
2353:
2347:
2343:
2335:
2332:
2331:
2283:
2280:
2279:
2259:
2255:
2237:
2233:
2215:
2211:
2202:
2198:
2190:
2187:
2186:
2144:
2141:
2140:
2106:
2103:
2102:
2095:
2087:Banach algebras
2063:
2051:
2044:
2043:
2037:
2031:
2023:
2017:
2008:
1979:
1967:
1961:
1955:
1949:
1931:
1928:
1922:
1919:
1913:
1910:
1904:
1901:
1895:
1889:
1886:
1880:
1877:
1871:
1865:
1854:
1850:
1839:
1838:
1832:
1829:
1823:
1757:
1727:Euclidean space
1723:
1698:
1697:
1688:
1684:
1677:
1662:
1649:
1644:
1637:
1622:
1607:
1603:
1597:
1590:
1575:
1574:
1555:
1541:
1531:
1524:
1510:
1493:
1488:
1484:
1471:
1456:
1455:
1446:
1442:
1435:
1421:
1408:
1397:
1395:
1388:
1374:
1355:
1351:
1350:
1337:
1333:
1332:
1330:
1326:
1313:
1298:
1297:
1288:
1284:
1277:
1263:
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1233:
1225:
1221:
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1206:
1190:
1188:
1185:
1184:
1152:
1148:
1140:
1137:
1136:
1132:
1128:
1118:
1108:
1097:
1093:
1089:
1062:
1051:
1049:
1032:
1029:
1028:
1007:
983:
979:
970:
966:
942:
938:
929:
925:
904:
900:
898:
895:
894:
887:
880:
866:
865:
846:
841:
834:
819:
815:
812:
811:
798:
791:
776:
772:
769:
768:
749:
734:
730:
726:
724:
721:
720:
712:
708:
698:
684:
673:
668:
591:
587:
586:
578:
576:
561:
557:
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529:
527:
524:
523:
520:
512:
494:
441:
438:
437:
414:
410:
380:
376:
367:
363:
361:
358:
357:
334:
330:
321:
317:
315:
312:
311:
293:
290:
267:
257:
251:
241:
213:complementation
185:
180:
166:
152:
134:
125:
119:
111:
90:
79:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3462:
3452:
3451:
3446:
3432:
3431:
3417:
3416:External links
3414:
3413:
3412:
3394:
3381:
3352:
3324:(1): 137–143.
3311:
3308:
3305:
3304:
3285:
3279:John S. Rose.
3272:
3253:
3236:
3220:
3197:
3145:
3117:
3110:
3090:
3084:
3064:
3042:
3041:
3039:
3036:
3035:
3034:
3029:
3024:
3019:
3012:
3009:
2997:(RGB)) = RGB,
2905:
2902:
2773:
2770:
2746:Coxeter groups
2676:transpositions
2598:
2595:
2594:
2593:
2590:
2554:
2551:
2511:
2510:
2499:
2494:
2490:
2486:
2483:
2480:
2475:
2471:
2467:
2464:
2461:
2458:
2453:
2449:
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2439:
2435:
2432:
2418:
2417:
2406:
2401:
2397:
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2390:
2387:
2382:
2378:
2374:
2371:
2368:
2365:
2360:
2356:
2350:
2346:
2342:
2339:
2329:
2328:(it is linear)
2317:
2314:
2311:
2308:
2305:
2302:
2299:
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2290:
2287:
2267:
2262:
2258:
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2245:
2240:
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2229:
2226:
2223:
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2210:
2205:
2201:
2197:
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2184:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
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2148:
2125:
2122:
2119:
2116:
2113:
2110:
2094:
2091:
1926:
1917:
1908:
1899:
1884:
1875:
1822:
1821:Linear algebra
1819:
1807:
1806:
1791:
1768:
1756:
1753:
1722:
1719:
1696:
1691:
1687:
1683:
1680:
1678:
1676:
1673:
1670:
1667:
1663:
1658:
1655:
1652:
1648:
1643:
1640:
1638:
1636:
1633:
1630:
1627:
1623:
1618:
1615:
1610:
1606:
1601:
1596:
1593:
1591:
1589:
1586:
1583:
1580:
1577:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1556:
1554:
1551:
1548:
1545:
1542:
1538:
1535:
1530:
1527:
1525:
1523:
1520:
1517:
1514:
1511:
1508:
1502:
1499:
1496:
1492:
1487:
1483:
1480:
1477:
1474:
1472:
1470:
1467:
1464:
1461:
1458:
1457:
1454:
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1411:
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1394:
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1378:
1375:
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1366:
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1358:
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1336:
1329:
1325:
1322:
1319:
1316:
1314:
1312:
1309:
1306:
1303:
1300:
1299:
1296:
1291:
1287:
1283:
1280:
1278:
1276:
1273:
1270:
1267:
1264:
1262:
1259:
1256:
1253:
1250:
1248:
1246:
1243:
1240:
1237:
1234:
1228:
1224:
1220:
1217:
1212:
1209:
1207:
1205:
1202:
1199:
1196:
1193:
1192:
1172:
1169:
1166:
1163:
1158:
1155:
1151:
1147:
1144:
1096:which satisfy
1074:
1071:
1068:
1065:
1060:
1057:
1054:
1048:
1045:
1042:
1039:
1036:
1014:
1011:
1006:
1003:
1000:
997:
994:
991:
986:
982:
978:
973:
969:
965:
962:
959:
956:
953:
950:
945:
941:
937:
932:
928:
924:
921:
918:
915:
912:
907:
903:
864:
858:
855:
852:
849:
845:
840:
837:
835:
833:
830:
827:
822:
818:
814:
813:
810:
805:
802:
797:
794:
792:
790:
787:
784:
779:
775:
771:
770:
767:
764:
761:
758:
755:
752:
750:
748:
745:
742:
737:
733:
729:
728:
672:
669:
667:
664:
646:have the same
629:
623:
620:
617:
614:
611:
608:
605:
602:
599:
594:
590:
584:
581:
573:
568:
565:
560:
555:
552:
549:
545:
541:
536:
532:
516:
509:Young tableaux
463:
462:
451:
448:
445:
423:
420:
417:
413:
409:
406:
403:
400:
397:
394:
389:
386:
383:
379:
375:
370:
366:
345:
342:
337:
333:
329:
324:
320:
297:= 0, 1, 2, ...
289:
286:
133:
130:
108:
107:
26:
18:Self-inversion
9:
6:
4:
3:
2:
3461:
3450:
3447:
3445:
3442:
3441:
3439:
3429:
3424:
3420:
3419:
3409:
3405:
3404:
3399:
3395:
3392:
3388:
3384:
3382:0-8218-0904-0
3378:
3374:
3370:
3366:
3362:
3358:
3353:
3349:
3345:
3341:
3337:
3332:
3327:
3323:
3319:
3314:
3313:
3300:
3296:
3289:
3282:
3276:
3270:
3267:, pp. 244–8,
3266:
3262:
3257:
3250:
3246:
3240:
3233:
3229:
3224:
3218:
3214:
3210:
3204:
3202:
3193:
3189:
3185:
3181:
3177:
3173:
3169:
3165:
3164:
3159:
3155:
3149:
3142:
3138:
3134:
3132:
3127:
3121:
3113:
3111:9780817649982
3107:
3103:
3102:
3094:
3087:
3085:9781440054167
3081:
3077:
3076:
3068:
3061:
3057:
3053:
3047:
3043:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3014:
3008:
3004:
3000:
2996:
2992:
2985:
2981:
2977:
2970:
2964:
2957:
2953:
2949:
2942:
2940:
2935:
2933:
2929:
2928:stream cipher
2925:
2920:
2918:
2914:
2911:
2901:
2899:
2895:
2891:
2887:
2883:
2879:
2874:
2872:
2868:
2864:
2860:
2856:
2852:
2848:
2847:Boolean logic
2844:
2840:
2836:
2831:
2829:
2825:
2821:
2817:
2813:
2809:
2804:
2801:
2795:
2789:
2788:
2783:
2779:
2769:
2767:
2763:
2758:
2752:
2747:
2743:
2740:
2736:
2732:
2728:
2724:
2718:
2714:
2708:
2703:
2702:strongly real
2698:
2692:
2686:
2684:
2679:
2677:
2673:
2668:
2666:
2662:
2658:
2657:
2652:
2648:
2643:
2637:
2633:
2627:
2623:
2617:
2612:
2608:
2604:
2591:
2589:
2585:
2584:complex plane
2581:
2578:
2577:
2576:
2574:
2570:
2566:
2560:
2550:
2548:
2544:
2540:
2534:
2530:
2526:
2520:
2516:
2492:
2488:
2481:
2473:
2469:
2462:
2459:
2451:
2447:
2441:
2437:
2430:
2423:
2422:
2421:
2399:
2395:
2388:
2380:
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2369:
2366:
2358:
2354:
2348:
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2337:
2330:
2312:
2306:
2303:
2300:
2294:
2291:
2285:
2260:
2256:
2249:
2246:
2238:
2234:
2227:
2224:
2216:
2212:
2208:
2203:
2199:
2192:
2185:
2170:
2167:
2158:
2152:
2146:
2139:
2138:
2137:
2120:
2114:
2108:
2100:
2090:
2088:
2084:
2080:
2076:
2071:
2069:
2061:
2057:
2054:
2047:
2040:
2034:
2030:
2026:
2020:
2016:
2011:
2006:
2001:
1999:
1995:
1991:
1987:
1982:
1978:
1973:
1970:
1964:
1958:
1952:
1946:
1942:
1938:
1934:
1925:
1916:
1907:
1898:
1892:
1883:
1874:
1868:
1862:
1860:
1847:
1842:
1835:
1828:
1818:
1817:of period 2.
1816:
1812:
1804:
1803:double points
1800:
1796:
1792:
1789:
1785:
1781:
1777:
1773:
1769:
1766:
1765:
1764:
1762:
1752:
1750:
1745:
1743:
1738:
1736:
1732:
1728:
1718:
1716:
1711:
1694:
1689:
1685:
1681:
1679:
1671:
1665:
1656:
1653:
1650:
1646:
1641:
1639:
1631:
1625:
1616:
1613:
1608:
1604:
1599:
1594:
1592:
1584:
1578:
1571:
1568:
1565:
1562:
1559:
1557:
1549:
1543:
1536:
1533:
1528:
1526:
1518:
1512:
1506:
1500:
1497:
1494:
1490:
1485:
1481:
1478:
1475:
1473:
1465:
1459:
1452:
1447:
1443:
1439:
1437:
1429:
1423:
1415:
1412:
1409:
1404:
1401:
1398:
1392:
1390:
1382:
1376:
1370:
1364:
1361:
1356:
1352:
1346:
1343:
1338:
1334:
1327:
1323:
1320:
1317:
1315:
1307:
1301:
1294:
1289:
1285:
1281:
1279:
1271:
1265:
1260:
1257:
1254:
1251:
1249:
1241:
1235:
1226:
1222:
1218:
1215:
1210:
1208:
1200:
1194:
1170:
1167:
1164:
1161:
1156:
1153:
1149:
1145:
1142:
1125:
1121:
1115:
1111:
1106:
1100:
1072:
1069:
1066:
1063:
1058:
1055:
1052:
1046:
1040:
1034:
1012:
1009:
1004:
1001:
995:
984:
980:
976:
971:
967:
960:
954:
943:
939:
935:
930:
926:
919:
913:
905:
901:
890:
883:
862:
856:
853:
850:
847:
843:
838:
836:
828:
820:
816:
808:
803:
800:
795:
793:
785:
777:
773:
765:
762:
759:
756:
753:
751:
743:
735:
731:
718:
705:
701:
696:
691:
687:
682:
678:
663:
661:
657:
653:
649:
645:
640:
627:
621:
615:
612:
609:
606:
600:
597:
592:
588:
582:
579:
566:
563:
553:
550:
547:
543:
539:
534:
530:
519:
515:
510:
506:
502:
497:
492:
488:
484:
480:
476:
472:
468:
449:
446:
443:
421:
418:
415:
411:
404:
401:
398:
392:
387:
384:
381:
377:
373:
368:
364:
343:
340:
335:
331:
327:
322:
318:
310:
309:
308:
306:
302:
296:
285:
282:
278:
274:
270:
265:
260:
254:
248:
244:
240:
235:
233:
230:
226:
222:
218:
214:
210:
206:
202:
198:
194:
188:
183:
178:
173:
169:
164:
163:reciprocation
159:
155:
150:
146:
141:
139:
129:
122:
117:
105:
101:
97:
93:
89:
88:
87:
85:
78:
74:
70:
66:
62:
53:
49:
45:
39:
33:
19:
3401:
3398:"Involution"
3368:
3361:Rost, Markus
3331:math/0506034
3321:
3317:
3298:
3288:
3275:
3264:
3256:
3244:
3239:
3231:
3223:
3167:
3161:
3157:
3148:
3129:
3120:
3100:
3093:
3074:
3067:
3051:
3046:
3022:Automorphism
3005:(BGR)) = BGR
3002:
2998:
2994:
2990:
2983:
2979:
2975:
2968:
2962:
2955:
2951:
2947:
2943:
2936:
2921:
2907:
2875:
2869:), IMTL and
2832:
2812:truth values
2807:
2805:
2799:
2793:
2785:
2775:
2756:
2750:
2744:
2738:
2734:
2730:
2726:
2722:
2720:(where
2716:
2712:
2706:
2696:
2690:
2687:
2680:
2669:
2664:
2660:
2654:
2650:
2641:
2635:
2631:
2625:
2621:
2615:
2603:group theory
2600:
2597:Group theory
2568:
2562:
2532:
2528:
2524:
2512:
2419:
2096:
2072:
2058:
2052:
2045:
2038:
2032:
2029:endomorphism
2024:
2018:
2009:
2002:
1980:
1974:
1968:
1962:
1956:
1950:
1944:
1940:
1936:
1932:
1923:
1914:
1912:, and sends
1905:
1896:
1890:
1881:
1872:
1866:
1863:
1858:
1845:
1840:
1833:
1830:
1810:
1808:
1802:
1783:
1779:
1761:projectivity
1758:
1746:
1739:
1724:
1712:
1126:
1119:
1113:
1109:
1098:
888:
881:
716:
703:
699:
694:
689:
685:
674:
641:
517:
513:
464:
294:
291:
280:
276:
272:
268:
258:
252:
246:
242:
236:
223:such as the
199:, half-turn
186:
181:
171:
167:
157:
153:
145:identity map
142:
135:
120:
109:
103:
99:
95:
91:
72:
68:
64:
58:
51:
47:
43:
3228:J. V. Field
3027:Idempotence
2924:bitwise NOT
2859:BL-algebras
2855:MV-algebras
2824:fuzzy logic
2694:of a group
2688:An element
2672:permutation
2567:, the word
2565:ring theory
2553:Ring theory
2083:C*-algebras
1960:. That is,
1894:that sends
1815:correlation
1795:fixed point
1183:, such as:
656:fixed point
239:composition
61:mathematics
3438:Categories
3428:Involution
3391:0955.16001
3170:(2): 144,
3154:Zagier, D.
3060:0321307143
3038:References
2808:involutive
2764:and their
2710:with
2700:is called
2665:involution
2619:such that
2569:involution
2068:correspond
1813:that is a
1784:Collection
1733:through a
1731:reflection
717:involution
652:odd number
493:(sequence
217:set theory
197:reflection
193:arithmetic
65:involution
3408:EMS Press
2932:keystream
2894:inclusion
2559:*-algebra
2547:transpose
2304:λ
2292:λ
2112:↦
1986:transpose
1776:Desargues
1654:−
1614:−
1566:
1498:
1482:
1413:−
1362:−
1324:
1258:−
1219:−
1168:∘
1162:∘
1154:−
1070:−
1005:−
977:∘
936:∘
854:−
760:−
681:symmetric
610:−
572:⌋
559:⌊
544:∑
419:−
402:−
385:−
307:in 1800:
303:found by
138:bijection
3367:(1998),
3348:45639619
3128:(1973),
3062:, p. 165
3054:, 2006,
3011:See also
2892:through
2782:negation
2639:, where
1948:for all
1855:−1
1811:polarity
229:Beaufort
209:geometry
201:rotation
149:negation
110:for all
77:function
46: :
3410:, 2001
3263:(1969)
3192:1041893
3184:2323918
3141:0445948
2890:ordered
2645:is the
2582:on the
2545:) with
2013:over a
2005:modules
1780:Porisms
695:general
499:in the
496:A000085
264:commute
175:), and
114:in the
84:inverse
3389:
3379:
3346:
3190:
3182:
3139:
3108:
3082:
3058:
3017:Atbash
2841:among
2519:groups
1977:matrix
1853:s and
648:parity
219:; and
203:, and
116:domain
3344:S2CID
3326:arXiv
3180:JSTOR
3032:ROT13
2917:masks
2661:group
2651:group
2611:order
2607:group
2527:) = (
2097:In a
2062:; if
2022:, an
1943:)) =
1735:plane
1107:with
893:then
711:with
677:graph
469:, 1,
225:ROT13
191:) in
102:)) =
75:is a
71:, or
63:, an
3377:ISBN
3215:via
3106:ISBN
3080:ISBN
3056:ISBN
2966:and
2908:The
2857:and
2629:and
2278:and
2081:and
2015:ring
1879:and
1101:≠ −1
1092:and
886:and
675:The
501:OEIS
447:>
436:for
356:and
256:and
237:The
170:↦ 1/
143:The
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3172:doi
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2601:In
2563:In
2521:as
2073:In
2036:of
1996:or
1954:in
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1903:to
1786:of
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1479:exp
1124:.)
1122:= 1
1112:= −
491:232
215:in
207:in
161:),
156:↦ −
118:of
59:In
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487:76
485:,
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481:,
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477:,
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2019:R
2010:M
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1969:V
1963:f
1957:V
1951:x
1945:x
1941:x
1939:(
1937:f
1935:(
1933:f
1927:1
1924:e
1918:2
1915:e
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