31:
3004:. For example, suppose the initial value of a variable is 3 and there is a subroutine sequence that reads the variable, then changes it to 5, and then reads it again. Each step in the sequence is idempotent: both steps reading the variable have no side effects and the step changing the variable to 5 will always have the same effect no matter how many times it is executed. Nonetheless, executing the entire sequence once produces the output (3, 5), but executing it a second time produces the output (5, 5), so the sequence is not idempotent.
3253:
3185:). Updating and deleting a given data are each usually idempotent as long as the request uniquely identifies the resource and only that resource again in the future. PUT and DELETE with unique identifiers reduce to the simple case of assignment to a variable of either a value or the null-value, respectively, and are idempotent for the same reason; the end result is always the same as the result of the initial execution, even if the response differs.
3189:
identifiers, so the creation of the identifier is delegated to the receiving system which then creates a corresponding new record. Similarly, PUT and DELETE requests with nonspecific criteria may result in different outcomes depending on the state of the system - for example, a request to delete the most recent record. In each case, subsequent executions will further modify the state of the system, so they are not idempotent.
3179:. Of the major HTTP methods, GET, PUT, and DELETE should be implemented in an idempotent manner according to the standard, but POST doesn't need to be. GET retrieves the state of a resource; PUT updates the state of a resource; and DELETE deletes a resource. As in the example above, reading data usually has no side effects, so it is idempotent (in fact
3268:. The initial activation of the button moves the system into a requesting state, until the request is satisfied. Subsequent activations of the button between the initial activation and the request being satisfied have no effect, unless the system is designed to adjust the time for satisfying the request based on the number of activations.
2995:
is typically idempotent, since this will not cause the database to change. Similarly, a request for changing a customer's address to XYZ is typically idempotent, because the final address will be the same no matter how many times the request is submitted. However, a customer request for placing an
3188:
Violation of the unique identification requirement in storage or deletion typically causes violation of idempotence. For example, storing or deleting a given set of content without specifying a unique identifier: POST requests, which do not need to be idempotent, often do not contain unique
2982:
This is a very useful property in many situations, as it means that an operation can be repeated or retried as often as necessary without causing unintended effects. With non-idempotent operations, the algorithm may have to keep track of whether the operation was already performed or not.
3947:
2958:
is idempotent if multiple calls to the subroutine have the same effect on the system state as a single call, in other words if the function from the system state space to itself associated with the subroutine is idempotent in the mathematical sense given in
3419:
The defining equation of nilpotent and idempotent expressions are respectively A = 0 and A = A; but with reference to idempotent expressions, it will always be assumed that they are of the form A = A unless it be otherwise distinctly stated."
2996:
order is typically not idempotent since multiple requests will lead to multiple orders being placed. A request for canceling a particular order is idempotent because no matter how many requests are made the order remains canceled.
2230:
2166:
2999:
A sequence of idempotent subroutines where at least one subroutine is different from the others, however, is not necessarily idempotent if a later subroutine in the sequence changes a value that an earlier subroutine depends
3794:
3228:
Many operations that are idempotent often have ways to "resume" a process if it is interrupted – ways that finish much faster than starting all over from the beginning. For example,
2607:
3211:
can load the page from disk and then simply re-execute the faulted instruction. In a processor where such instructions are not idempotent, dealing with page faults is much more complex.
1119:
1074:
2901:
1406:
1333:
464:
4081:
181:
in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from
3719:
1498:
1155:
975:
2824:
2316:
620:
1454:
4016:
2721:
1930:
1850:
2019:
1761:
1731:
1659:
1693:
579:
545:
1964:
935:
842:
803:
769:
690:
390:
322:
2855:
1621:
1367:
1294:
3787:
2776:
2748:
2528:
1588:
1548:
2667:
2101:
2045:
1897:
416:
350:
286:
262:
2797:
1568:
1525:
1261:
1241:
1221:
1198:
2075:
2474:
1002:
154:
whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in
4129:
4109:
3760:
3740:
3680:
3660:
2495:
2447:
2427:
2407:
2250:
2125:
1870:
1175:
1026:
902:
882:
862:
736:
713:
648:
512:
492:
239:
219:
3225:(SOA), a multiple-step orchestration process composed entirely of idempotent steps can be replayed without side-effects if any part of that process fails.
2173:
2139:
4499:
3633:
This is an equation between functions. Two functions are equal if their domains and ranges agree, and their output values agree on their whole domain.
2627:
4494:
4197:
It knows that repeating the request will have the same intended effect, even if the original request succeeded, though the response might differ.
3196:, idempotence refers to the ability of a system to produce the same outcome, even if the same file, event or message is received more than once.
4514:
4468:
4426:
4397:
4363:
4289:
3601:
4274:
3461:
3218:
is expected to be idempotent. In other words, if the output is already "pretty", there should be nothing to do for the pretty-printer.
4257:
For example, this design specification includes detailed algorithm for when elevator cars will respond to subsequent calls for service
2633:
Neither the property of being idempotent nor that of being not is preserved under function composition. As an example for the former,
3509:
3392:
3942:{\displaystyle (f\circ g)\circ (f\circ g)=f\circ (g\circ f)\circ g=f\circ (f\circ g)\circ g=(f\circ f)\circ (g\circ g)=f\circ g}
49:
button (green) is an idempotent operation, since it has the same effect whether done once or multiple times. Likewise, pressing
4509:
4225:
2955:
4343:
4238:
3222:
2536:
3405:
Original manuscript of 1870 lecture before
National Academy of Sciences (Washington, DC, USA): Peirce, Benjamin (1870)
3200:
4338:
4156:
3176:
3172:
3168:
2382:
2274:
2270:
3409:
From pages 16-17: "When an expression which is raised to the square or any higher power vanishes, it may be called
4504:
4318:
4212:
159:
3367:
3287:
2326:
2104:
1201:
4460:
4454:
4148:
3485:
3343:
3318:
3313:
2929:
2917:
1079:
1034:
171:
20:
2862:
30:
4358:, Mathematics and its Applications, vol. 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380,
3193:
1764:
1372:
1299:
435:
143:
4021:
2967:
2947:
2921:
1873:
167:
4384:, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385,
1768:
3686:
3252:
1459:
1127:
942:
2803:
2287:
591:
4189:
3292:
2530:
is the number of different idempotent functions. Hence, taking into account all possible partitions,
1877:
1415:
3962:
2676:
2620:= 0, 1, 2, 3, 4, 5, 6, 7, 8, ... starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, ... (sequence
1903:
1816:
4351:
4132:
3426:
3307:
3297:
2352:
1528:
1505:
810:
1971:
1744:
1698:
1626:
3446:
2670:
2348:
1666:
1178:
552:
518:
4333:
1937:
908:
815:
776:
742:
663:
363:
295:
2831:
1594:
1340:
1267:
4464:
4422:
4393:
4359:
4285:
4169:
3766:
3597:
3505:
3413:; but when raised to a square or higher power it gives itself as the result, it may be called
3323:
3302:
2905:
2755:
2727:
2356:
2263:
2256:
1797:
1782:
716:
24:
4444:
3406:
2500:
1573:
1533:
4432:
4385:
4322:
4179:
3438:
3328:
3282:
3265:
3208:
2935:
2637:
2613:
2080:
2024:
1882:
693:
395:
335:
271:
247:
163:
155:
151:
119:
113:
69:
42:
4478:
4407:
4373:
4307:
4299:
2782:
1553:
1510:
1246:
1226:
1206:
1183:
4474:
4436:
4403:
4369:
4326:
4295:
4208:
4152:
3387:
3241:
2278:
2051:
1801:
1005:
658:
242:
178:
63:
2453:
981:
3180:
4114:
4094:
3745:
3725:
3665:
3645:
3590:
3382:
3237:
2480:
2432:
2412:
2392:
2319:
2235:
2132:
2110:
1855:
1501:
1160:
1011:
887:
867:
847:
721:
698:
633:
623:
497:
477:
471:
467:
224:
204:
4284:(2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412,
4488:
4226:"Compiler construction of idempotent regions and applications in architecture design"
3338:
3277:
2971:
2225:{\displaystyle \operatorname {abs} (\operatorname {abs} (x))=\operatorname {abs} (x)}
3260:
Applied examples that many people could encounter in their day-to-day lives include
4145:
2378:
2374:
2337:
2161:{\displaystyle \operatorname {abs} \circ \operatorname {abs} =\operatorname {abs} }
1775:
4239:"Geared Traction Passenger Elevator Specification Guide Information/Instructions"
3545:
3499:
4192:
4173:
3215:
2925:
2367:
2363:
2341:
2333:
1793:
1789:
147:
4459:, Algebras and Applications, vol. 1, Kluwer Academic Publishers, pp.
4249:
4414:
4389:
3204:
2951:
2942:
may have a different meaning depending on the context in which it is applied:
1738:
3333:
1122:
189:
183:
3261:
2992:
1800:
is either 0 or 1. If the determinant is 1, the matrix necessarily is the
627:
3450:
3362:
3310: – a generalization of idempotence to binary relations
4184:
3229:
2974:
is idempotent if it is idempotent in the mathematical sense given in
2612:
is the total number of possible idempotent functions on the set. The
586:
430:
3442:
2616:
of the number of idempotent functions as given by the sum above for
3622:. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc.
2779:
happens to be. As an example for the latter, the negation function
3251:
3233:
29:
4315:
Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994
3256:
A typical crosswalk button is an example of an idempotent system
2370:
functions of the power set of a monoid to itself are idempotent;
4270:
3240:, installing an application and all of its dependencies with a
2329:
function from the power set of a group to itself is idempotent;
99:
4175:
Hypertext
Transfer Protocol (HTTP/1.1): Semantics and Content
4146:
Hypertext
Transfer Protocol (HTTP/1.1): Semantics and Content
3504:. Berlin: Springer Science & Business Media. p. 22.
1384:
1311:
1133:
1088:
1043:
2622:
131:
125:
93:
81:
3556:
un magma, noté multiplicativement. On nomme idempotent de
2991:
A function looking up a customer's name and address in a
75:
3501:
Linear
Algebra: An Introduction to Abstract Mathematics
3469:. New York, New York, USA: D. Van Nostrand. p. 8.
3002:
idempotence is not closed under sequential composition
4117:
4097:
4024:
3965:
3797:
3769:
3748:
3728:
3689:
3668:
3648:
2865:
2834:
2806:
2785:
2758:
2730:
2679:
2640:
2539:
2503:
2483:
2456:
2435:
2415:
2395:
2290:
2238:
2176:
2142:
2113:
2083:
2054:
2027:
1974:
1940:
1906:
1885:
1858:
1819:
1747:
1701:
1669:
1629:
1597:
1576:
1556:
1536:
1513:
1462:
1418:
1375:
1343:
1302:
1270:
1249:
1229:
1209:
1186:
1163:
1130:
1082:
1037:
1014:
984:
945:
911:
890:
870:
850:
818:
779:
745:
724:
701:
666:
636:
594:
555:
521:
500:
480:
438:
398:
366:
338:
298:
274:
250:
227:
207:
102:
72:
4453:
Polcino Milies, César; Sehgal, Sudarshan K. (2002),
128:
96:
90:
78:
2602:{\displaystyle \sum _{k=0}^{n}{n \choose k}k^{n-k}}
134:
122:
87:
84:
4421:(Third ed.), Reading, Mass.: Addison-Wesley,
4123:
4103:
4075:
4010:
3941:
3781:
3754:
3734:
3713:
3674:
3654:
3589:
2895:
2849:
2818:
2791:
2770:
2742:
2715:
2661:
2601:
2522:
2489:
2468:
2441:
2421:
2401:
2310:
2244:
2224:
2160:
2119:
2095:
2069:
2039:
2013:
1958:
1924:
1891:
1864:
1844:
1755:
1725:
1687:
1653:
1615:
1582:
1562:
1542:
1519:
1492:
1448:
1400:
1361:
1327:
1288:
1255:
1235:
1215:
1192:
1169:
1149:
1113:
1068:
1020:
996:
969:
929:
896:
876:
856:
836:
797:
763:
730:
707:
684:
642:
614:
573:
539:
506:
486:
458:
410:
384:
344:
316:
280:
256:
233:
213:
177:The term was introduced by American mathematician
2904:is. In both cases, the composition is simply the
2577:
2564:
2695:
3207:are idempotent. So if a page fault occurs, the
4354:; Gubareni, Nadiya; Kirichenko, V. V. (2004),
3371:(3rd ed.). Oxford University Press. 2010.
3481:
2799:on the Boolean domain is not idempotent, but
170:(in which it is connected to the property of
8:
1720:
1708:
1648:
1636:
1478:
1466:
1434:
1422:
3203:, instructions that might possibly cause a
864:is the only idempotent element. Indeed, if
3550:(in French). Paris: Vuibert. p. 180.
4183:
4116:
4096:
4023:
3964:
3950:, using the associativity of composition.
3796:
3768:
3747:
3727:
3688:
3667:
3647:
2864:
2833:
2805:
2784:
2757:
2729:
2678:
2639:
2587:
2576:
2563:
2561:
2555:
2544:
2538:
2508:
2502:
2482:
2455:
2434:
2414:
2394:
2291:
2289:
2237:
2175:
2141:
2112:
2082:
2053:
2026:
1973:
1939:
1905:
1884:
1857:
1827:
1818:
1749:
1748:
1746:
1700:
1668:
1628:
1596:
1575:
1555:
1535:
1512:
1461:
1417:
1383:
1382:
1374:
1342:
1310:
1309:
1301:
1269:
1248:
1228:
1208:
1185:
1162:
1132:
1131:
1129:
1114:{\displaystyle ({\mathcal {P}}(E),\cap )}
1087:
1086:
1081:
1069:{\displaystyle ({\mathcal {P}}(E),\cup )}
1042:
1041:
1036:
1013:
983:
944:
910:
889:
869:
849:
817:
778:
744:
723:
700:
665:
635:
599:
598:
593:
554:
520:
499:
479:
443:
442:
437:
397:
365:
337:
297:
273:
249:
226:
206:
2896:{\displaystyle -(\cdot )\circ -(\cdot )}
2858:of real numbers is not idempotent, but
1899:, idempotent elements are the functions
4246:NC Department Of Labor, Elevator Bureau
3354:
3175:are the major attributes that separate
738:, if it exists, is idempotent. Indeed,
4382:A first course in noncommutative rings
1401:{\displaystyle x\in {\mathcal {P}}(E)}
1328:{\displaystyle x\in {\mathcal {P}}(E)}
459:{\displaystyle (\mathbb {N} ,\times )}
7:
4275:Free On-line Dictionary of Computing
4076:{\displaystyle f(g(1))=f(5)=2\neq 1}
4356:Algebras, rings and modules. vol. 1
3682:commute under composition (i.e. if
2429:elements, we can partition it into
2813:
2807:
2786:
2568:
2336:function from the power set of an
2295:
2292:
1004:by multiplying on the left by the
14:
4091:also showing that commutation of
3714:{\displaystyle f\circ g=g\circ f}
1493:{\displaystyle (\{0,1\},\wedge )}
1150:{\displaystyle {\mathcal {P}}(E)}
970:{\displaystyle x\cdot x=x\cdot e}
158:(in particular, in the theory of
19:For the concepts in algebra, see
4500:Algebraic properties of elements
4313:, in Gunawardena, Jeremy (ed.),
4308:"An introduction to idempotency"
3395:from the original on 2016-10-19.
2819:{\displaystyle \neg \circ \neg }
2355:functions of the power set of a
2311:{\displaystyle \mathrm {Re} (z)}
615:{\displaystyle (\mathbb {N} ,+)}
118:
68:
4495:Properties of binary operations
3431:American Journal of Mathematics
2975:
2960:
1778:, multiplication is idempotent.
1449:{\displaystyle (\{0,1\},\vee )}
4456:An Introduction to Group Rings
4058:
4052:
4043:
4040:
4034:
4028:
4011:{\displaystyle f(g(7))=f(7)=1}
3999:
3993:
3984:
3981:
3975:
3969:
3924:
3912:
3906:
3894:
3882:
3870:
3852:
3840:
3828:
3816:
3810:
3798:
2890:
2884:
2875:
2869:
2844:
2838:
2827:is. Similarly, unary negation
2716:{\displaystyle g(x)=\max(x,5)}
2710:
2698:
2689:
2683:
2650:
2644:
2305:
2299:
2219:
2213:
2201:
2198:
2192:
2183:
2064:
2058:
2008:
2002:
1993:
1990:
1984:
1978:
1925:{\displaystyle f\colon E\to E}
1916:
1845:{\displaystyle (E^{E},\circ )}
1839:
1820:
1487:
1463:
1443:
1419:
1395:
1389:
1322:
1316:
1144:
1138:
1108:
1099:
1093:
1083:
1063:
1054:
1048:
1038:
831:
819:
679:
667:
609:
595:
453:
439:
1:
3547:Polynômes et algèbre linéaire
3223:service-oriented architecture
142:) is the property of certain
4515:Theoretical computer science
4306:Gunawardena, Jeremy (1998),
4135:for idempotency preservation
3427:"Linear associative algebra"
3407:"Linear associative algebra"
3150:// prints "5\n5\n"
3141:// prints "3\n5\n"
2014:{\displaystyle f(f(x))=f(x)}
1852:of the functions from a set
1756:{\displaystyle \mathbb {Z} }
1726:{\displaystyle x\in \{0,1\}}
1654:{\displaystyle x\in \{0,1\}}
45:control panel. Pressing the
4339:Encyclopedia of Mathematics
4157:HyperText Transfer Protocol
3722:) then idempotency of both
3169:Hypertext Transfer Protocol
2048:(in other words, the image
1688:{\displaystyle x\wedge x=x}
574:{\displaystyle 1\times 1=1}
540:{\displaystyle 0\times 0=0}
4531:
4446:Linear Associative Algebra
4319:Cambridge University Press
3463:Linear Associative Algebra
3214:When reformatting output,
2915:
1959:{\displaystyle f\circ f=f}
1790:ring of quadratic matrices
930:{\displaystyle x\cdot x=x}
837:{\displaystyle (G,\cdot )}
798:{\displaystyle a\cdot a=a}
764:{\displaystyle e\cdot e=e}
685:{\displaystyle (M,\cdot )}
385:{\displaystyle x\cdot x=x}
317:{\displaystyle x\cdot x=x}
18:
4390:10.1007/978-1-4419-8616-0
4282:von Neumann regular rings
3618:Garrett Birkhoff (1967).
3482:Polcino & Sehgal 2002
3460:Peirce, Benjamin (1882).
3425:Peirce, Benjamin (1881).
3368:Oxford English Dictionary
3288:Fixed point (mathematics)
2987:Computer science examples
2850:{\displaystyle -(\cdot )}
2723:are both idempotent, but
2359:to itself are idempotent;
2281:functions are idempotent;
2259:functions are idempotent;
1785:, addition is idempotent.
1616:{\displaystyle x\vee x=x}
1362:{\displaystyle x\cap x=x}
1289:{\displaystyle x\cup x=x}
4280:Goodearl, K. R. (1991),
3782:{\displaystyle f\circ g}
3544:Doneddu, Alfred (1976).
3498:Valenza, Robert (2012).
3344:Referential transparency
3319:Involution (mathematics)
3314:Idempotent (ring theory)
3230:resuming a file transfer
3171:(HTTP), idempotence and
3006:
2930:Command query separation
2918:Referential transparency
2912:Computer science meaning
2771:{\displaystyle g\circ f}
2743:{\displaystyle f\circ g}
2449:chosen fixed points and
2344:to itself is idempotent;
1590:are idempotent. Indeed,
1263:are idempotent. Indeed,
514:are idempotent. Indeed,
172:referential transparency
21:Idempotent (ring theory)
3588:George Grätzer (2003).
3201:load–store architecture
3194:event stream processing
2908:, which is idempotent.
2523:{\displaystyle k^{n-k}}
2477:non-fixed points under
2284:the real part function
2135:is idempotent. Indeed,
1583:{\displaystyle \wedge }
1543:{\displaystyle \wedge }
844:, the identity element
650:is idempotent. Indeed,
4510:Mathematical relations
4248:. 2002. Archived from
4125:
4105:
4077:
4012:
3943:
3783:
3756:
3736:
3715:
3676:
3656:
3624:. Here: Sect.I.5, p.8.
3592:General Lattice Theory
3257:
2968:functional programming
2948:imperative programming
2922:Reentrant (subroutine)
2897:
2851:
2820:
2793:
2772:
2744:
2717:
2663:
2662:{\displaystyle f(x)=x}
2603:
2560:
2524:
2491:
2470:
2443:
2423:
2403:
2312:
2246:
2226:
2162:
2121:
2097:
2096:{\displaystyle x\in E}
2071:
2041:
2040:{\displaystyle x\in E}
2015:
1960:
1926:
1893:
1892:{\displaystyle \circ }
1866:
1846:
1757:
1727:
1689:
1655:
1617:
1584:
1564:
1544:
1521:
1494:
1450:
1402:
1363:
1329:
1290:
1257:
1237:
1217:
1194:
1171:
1151:
1115:
1070:
1022:
998:
971:
931:
898:
878:
858:
838:
799:
765:
732:
709:
686:
644:
616:
575:
541:
508:
488:
460:
412:
411:{\displaystyle x\in S}
386:
346:
345:{\displaystyle \cdot }
318:
282:
281:{\displaystyle \cdot }
258:
257:{\displaystyle \cdot }
235:
215:
182:
168:functional programming
54:
16:Property of operations
4126:
4106:
4078:
4013:
3944:
3784:
3757:
3737:
3716:
3677:
3657:
3596:. Basel: Birkhäuser.
3520:of a magma such that
3255:
2898:
2852:
2821:
2794:
2792:{\displaystyle \neg }
2773:
2745:
2718:
2664:
2604:
2540:
2525:
2492:
2471:
2444:
2424:
2404:
2313:
2247:
2227:
2163:
2122:
2098:
2072:
2042:
2016:
1961:
1927:
1894:
1867:
1847:
1763:), the operations of
1758:
1728:
1690:
1656:
1618:
1585:
1565:
1563:{\displaystyle \vee }
1545:
1522:
1520:{\displaystyle \vee }
1495:
1451:
1403:
1364:
1330:
1291:
1258:
1256:{\displaystyle \cap }
1238:
1236:{\displaystyle \cup }
1218:
1216:{\displaystyle \cap }
1195:
1193:{\displaystyle \cup }
1172:
1152:
1116:
1071:
1023:
999:
972:
932:
899:
879:
859:
839:
800:
766:
733:
710:
687:
645:
617:
576:
542:
509:
489:
461:
413:
387:
347:
319:
283:
259:
236:
216:
41:buttons of a train's
33:
4170:"Idempotent Methods"
4115:
4095:
4022:
3963:
3795:
3767:
3746:
3726:
3687:
3666:
3646:
3608:Here: Sect.1.2, p.5.
3293:Idempotent of a code
2863:
2832:
2804:
2783:
2756:
2728:
2677:
2638:
2537:
2501:
2481:
2454:
2433:
2413:
2393:
2288:
2236:
2174:
2140:
2111:
2081:
2070:{\displaystyle f(x)}
2052:
2025:
1972:
1967:, that is such that
1938:
1904:
1883:
1878:function composition
1856:
1817:
1809:Idempotent functions
1745:
1699:
1667:
1627:
1595:
1574:
1554:
1534:
1511:
1460:
1416:
1373:
1341:
1300:
1268:
1247:
1227:
1207:
1184:
1161:
1128:
1080:
1035:
1012:
982:
943:
909:
888:
868:
848:
816:
777:
743:
722:
699:
664:
634:
592:
553:
519:
498:
478:
436:
396:
364:
336:
296:
272:
248:
225:
205:
4380:Lam, T. Y. (2001),
4352:Hazewinkel, Michiel
4224:Marc A. de Kruijf.
4178:. sec. 4.2.2.
4133:necessary condition
3308:Idempotent relation
3298:Idempotent analysis
3234:synchronizing files
2469:{\displaystyle n-k}
1529:logical conjunction
1506:logical disjunction
997:{\displaystyle x=e}
4443:Peirce, Benjamin.
4151:2014-06-08 at the
4121:
4101:
4073:
4008:
3939:
3779:
3752:
3732:
3711:
3672:
3652:
3258:
2893:
2847:
2816:
2789:
2768:
2740:
2713:
2659:
2599:
2520:
2487:
2466:
2439:
2419:
2399:
2327:subgroup generated
2308:
2242:
2222:
2158:
2117:
2093:
2067:
2037:
2011:
1956:
1922:
1889:
1874:set exponentiation
1862:
1842:
1753:
1723:
1685:
1651:
1613:
1580:
1560:
1540:
1517:
1490:
1446:
1398:
1359:
1325:
1286:
1253:
1233:
1213:
1190:
1167:
1147:
1111:
1066:
1018:
994:
967:
927:
894:
874:
854:
834:
795:
761:
728:
705:
682:
640:
612:
571:
537:
504:
484:
456:
408:
382:
342:
314:
278:
254:
231:
211:
55:
4505:Closure operators
4470:978-1-4020-0238-0
4428:978-0-201-55540-0
4399:978-0-387-95183-6
4365:978-1-4020-2690-4
4321:, pp. 1–49,
4291:978-0-89464-632-4
4124:{\displaystyle g}
4104:{\displaystyle f}
3755:{\displaystyle g}
3735:{\displaystyle f}
3675:{\displaystyle g}
3655:{\displaystyle f}
3603:978-3-7643-6996-5
3324:Iterated function
3303:Idempotent matrix
3266:crosswalk buttons
3264:call buttons and
2906:identity function
2751:is not, although
2575:
2490:{\displaystyle f}
2442:{\displaystyle k}
2422:{\displaystyle n}
2402:{\displaystyle E}
2357:topological space
2264:identity function
2245:{\displaystyle x}
2120:{\displaystyle f}
1865:{\displaystyle E}
1798:idempotent matrix
1783:Tropical semiring
1741:(for instance in
1170:{\displaystyle E}
1021:{\displaystyle x}
897:{\displaystyle G}
884:is an element of
877:{\displaystyle x}
857:{\displaystyle e}
731:{\displaystyle a}
717:absorbing element
708:{\displaystyle e}
643:{\displaystyle 0}
507:{\displaystyle 1}
487:{\displaystyle 0}
234:{\displaystyle S}
214:{\displaystyle x}
164:closure operators
25:Idempotent matrix
4522:
4481:
4439:
4410:
4376:
4347:
4329:
4312:
4302:
4258:
4256:
4254:
4243:
4235:
4229:
4222:
4216:
4206:
4200:
4199:
4187:
4185:10.17487/RFC7231
4166:
4160:
4142:
4136:
4130:
4128:
4127:
4122:
4110:
4108:
4107:
4102:
4089:
4083:
4082:
4080:
4079:
4074:
4017:
4015:
4014:
4009:
3957:
3951:
3949:
3948:
3946:
3945:
3940:
3789:
3788:
3786:
3785:
3780:
3762:implies that of
3761:
3759:
3758:
3753:
3741:
3739:
3738:
3733:
3721:
3720:
3718:
3717:
3712:
3681:
3679:
3678:
3673:
3661:
3659:
3658:
3653:
3640:
3634:
3631:
3625:
3623:
3615:
3609:
3607:
3595:
3585:
3579:
3578:
3541:
3535:
3534:
3495:
3489:
3479:
3473:
3470:
3468:
3454:
3403:
3397:
3396:
3379:
3373:
3372:
3359:
3329:List of matrices
3283:Closure operator
3248:Applied examples
3209:operating system
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
2936:computer science
2903:
2902:
2900:
2899:
2894:
2857:
2856:
2854:
2853:
2848:
2826:
2825:
2823:
2822:
2817:
2798:
2796:
2795:
2790:
2778:
2777:
2775:
2774:
2769:
2750:
2749:
2747:
2746:
2741:
2722:
2720:
2719:
2714:
2669:
2668:
2666:
2665:
2660:
2625:
2614:integer sequence
2608:
2606:
2605:
2600:
2598:
2597:
2582:
2581:
2580:
2567:
2559:
2554:
2529:
2527:
2526:
2521:
2519:
2518:
2496:
2494:
2493:
2488:
2476:
2475:
2473:
2472:
2467:
2448:
2446:
2445:
2440:
2428:
2426:
2425:
2420:
2408:
2406:
2405:
2400:
2322:, is idempotent.
2317:
2315:
2314:
2309:
2298:
2252:
2251:
2249:
2248:
2243:
2231:
2229:
2228:
2223:
2168:
2167:
2165:
2164:
2159:
2127:). For example:
2126:
2124:
2123:
2118:
2102:
2100:
2099:
2094:
2077:of each element
2076:
2074:
2073:
2068:
2047:
2046:
2044:
2043:
2038:
2020:
2018:
2017:
2012:
1966:
1965:
1963:
1962:
1957:
1932:
1931:
1929:
1928:
1923:
1898:
1896:
1895:
1890:
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1832:
1831:
1762:
1760:
1759:
1754:
1752:
1733:
1732:
1730:
1729:
1724:
1694:
1692:
1691:
1686:
1661:
1660:
1658:
1657:
1652:
1622:
1620:
1619:
1614:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1549:
1547:
1546:
1541:
1526:
1524:
1523:
1518:
1499:
1497:
1496:
1491:
1455:
1453:
1452:
1447:
1408:
1407:
1405:
1404:
1399:
1388:
1387:
1368:
1366:
1365:
1360:
1335:
1334:
1332:
1331:
1326:
1315:
1314:
1295:
1293:
1292:
1287:
1262:
1260:
1259:
1254:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1202:set intersection
1199:
1197:
1196:
1191:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1137:
1136:
1120:
1118:
1117:
1112:
1092:
1091:
1075:
1073:
1072:
1067:
1047:
1046:
1027:
1025:
1024:
1019:
1003:
1001:
1000:
995:
977:
976:
974:
973:
968:
937:
936:
934:
933:
928:
903:
901:
900:
895:
883:
881:
880:
875:
863:
861:
860:
855:
843:
841:
840:
835:
805:
804:
802:
801:
796:
771:
770:
768:
767:
762:
737:
735:
734:
729:
714:
712:
711:
706:
694:identity element
691:
689:
688:
683:
653:
649:
647:
646:
641:
621:
619:
618:
613:
602:
581:
580:
578:
577:
572:
547:
546:
544:
543:
538:
513:
511:
510:
505:
493:
491:
490:
485:
465:
463:
462:
457:
446:
418:
417:
415:
414:
409:
391:
389:
388:
383:
351:
349:
348:
343:
331:binary operation
324:
323:
321:
320:
315:
287:
285:
284:
279:
263:
261:
260:
255:
241:equipped with a
240:
238:
237:
232:
220:
218:
217:
212:
193:(same + power).
156:abstract algebra
152:computer science
141:
140:
137:
136:
133:
130:
127:
124:
117:
109:
108:
105:
104:
101:
98:
95:
92:
89:
86:
83:
80:
77:
74:
67:
43:destination sign
4530:
4529:
4525:
4524:
4523:
4521:
4520:
4519:
4485:
4484:
4471:
4452:
4429:
4413:
4400:
4379:
4366:
4350:
4332:
4310:
4305:
4292:
4279:
4266:
4264:Further reading
4261:
4252:
4241:
4237:
4236:
4232:
4223:
4219:
4213:"Demand Paging"
4209:John Ousterhout
4207:
4203:
4168:
4167:
4163:
4153:Wayback Machine
4143:
4139:
4113:
4112:
4093:
4092:
4090:
4086:
4020:
4019:
3961:
3960:
3958:
3954:
3793:
3792:
3791:
3765:
3764:
3763:
3744:
3743:
3724:
3723:
3685:
3684:
3683:
3664:
3663:
3644:
3643:
3641:
3637:
3632:
3628:
3617:
3616:
3612:
3604:
3587:
3586:
3582:
3543:
3542:
3538:
3512:
3497:
3496:
3492:
3480:
3476:
3466:
3459:
3443:10.2307/2369153
3424:
3418:
3404:
3400:
3388:Merriam-Webster
3381:
3380:
3376:
3361:
3360:
3356:
3352:
3274:
3250:
3242:package manager
3216:pretty-printing
3165:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
2989:
2932:
2914:
2861:
2860:
2859:
2830:
2829:
2828:
2802:
2801:
2800:
2781:
2780:
2754:
2753:
2752:
2726:
2725:
2724:
2675:
2674:
2636:
2635:
2634:
2621:
2583:
2562:
2535:
2534:
2504:
2499:
2498:
2479:
2478:
2452:
2451:
2450:
2431:
2430:
2411:
2410:
2391:
2390:
2373:the idempotent
2286:
2285:
2279:fractional part
2234:
2233:
2172:
2171:
2170:
2138:
2137:
2136:
2109:
2108:
2079:
2078:
2050:
2049:
2023:
2022:
1970:
1969:
1968:
1936:
1935:
1934:
1902:
1901:
1900:
1881:
1880:
1872:to itself (see
1854:
1853:
1823:
1815:
1814:
1811:
1802:identity matrix
1771:are idempotent.
1743:
1742:
1697:
1696:
1665:
1664:
1663:
1625:
1624:
1593:
1592:
1591:
1572:
1571:
1552:
1551:
1532:
1531:
1509:
1508:
1458:
1457:
1414:
1413:
1412:In the monoids
1371:
1370:
1339:
1338:
1337:
1298:
1297:
1266:
1265:
1264:
1245:
1244:
1225:
1224:
1205:
1204:
1182:
1181:
1159:
1158:
1126:
1125:
1078:
1077:
1033:
1032:
1031:In the monoids
1010:
1009:
1006:inverse element
980:
979:
941:
940:
939:
907:
906:
905:
886:
885:
866:
865:
846:
845:
814:
813:
775:
774:
773:
741:
740:
739:
720:
719:
697:
696:
662:
661:
651:
632:
631:
624:natural numbers
590:
589:
551:
550:
549:
517:
516:
515:
496:
495:
476:
475:
468:natural numbers
434:
433:
426:
394:
393:
362:
361:
360:
334:
333:
294:
293:
292:
270:
269:
246:
245:
243:binary operator
223:
222:
203:
202:
199:
179:Benjamin Peirce
121:
112:
111:
71:
62:
61:
28:
17:
12:
11:
5:
4528:
4526:
4518:
4517:
4512:
4507:
4502:
4497:
4487:
4486:
4483:
4482:
4469:
4450:
4441:
4427:
4411:
4398:
4377:
4364:
4348:
4330:
4303:
4290:
4277:
4265:
4262:
4260:
4259:
4255:on 2011-05-23.
4230:
4228:. 2012. p. 10.
4217:
4201:
4161:
4137:
4120:
4100:
4084:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3952:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3778:
3775:
3772:
3751:
3731:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3671:
3651:
3635:
3626:
3620:Lattice Theory
3610:
3602:
3580:
3536:
3510:
3490:
3474:
3472:
3471:
3456:
3398:
3374:
3353:
3351:
3348:
3347:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3305:
3300:
3295:
3290:
3285:
3280:
3273:
3270:
3249:
3246:
3238:software build
3236:, creating a
3007:
2988:
2985:
2980:
2979:
2976:the definition
2964:
2961:the definition
2913:
2910:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2846:
2843:
2840:
2837:
2815:
2812:
2809:
2788:
2767:
2764:
2761:
2739:
2736:
2733:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2658:
2655:
2652:
2649:
2646:
2643:
2610:
2609:
2596:
2593:
2590:
2586:
2579:
2574:
2571:
2566:
2558:
2553:
2550:
2547:
2543:
2517:
2514:
2511:
2507:
2486:
2465:
2462:
2459:
2438:
2418:
2398:
2387:
2386:
2371:
2360:
2345:
2330:
2323:
2320:complex number
2307:
2304:
2301:
2297:
2294:
2282:
2267:
2266:is idempotent;
2260:
2254:
2241:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2157:
2154:
2151:
2148:
2145:
2133:absolute value
2116:
2092:
2089:
2086:
2066:
2063:
2060:
2057:
2036:
2033:
2030:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1955:
1952:
1949:
1946:
1943:
1921:
1918:
1915:
1912:
1909:
1888:
1861:
1841:
1838:
1835:
1830:
1826:
1822:
1813:In the monoid
1810:
1807:
1806:
1805:
1786:
1779:
1772:
1751:
1735:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1684:
1681:
1678:
1675:
1672:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1612:
1609:
1606:
1603:
1600:
1579:
1559:
1550:respectively,
1539:
1516:
1502:Boolean domain
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1410:
1397:
1394:
1391:
1386:
1381:
1378:
1358:
1355:
1352:
1349:
1346:
1324:
1321:
1318:
1313:
1308:
1305:
1285:
1282:
1279:
1276:
1273:
1252:
1232:
1223:respectively,
1212:
1189:
1166:
1146:
1143:
1140:
1135:
1110:
1107:
1104:
1101:
1098:
1095:
1090:
1085:
1065:
1062:
1059:
1056:
1053:
1050:
1045:
1040:
1029:
1017:
993:
990:
987:
966:
963:
960:
957:
954:
951:
948:
926:
923:
920:
917:
914:
893:
873:
853:
833:
830:
827:
824:
821:
807:
794:
791:
788:
785:
782:
760:
757:
754:
751:
748:
727:
704:
681:
678:
675:
672:
669:
655:
639:
611:
608:
605:
601:
597:
583:
570:
567:
564:
561:
558:
536:
533:
530:
527:
524:
503:
483:
472:multiplication
455:
452:
449:
445:
441:
425:
422:
421:
420:
407:
404:
401:
381:
378:
375:
372:
369:
352:is said to be
341:
327:
326:
313:
310:
307:
304:
301:
277:
264:is said to be
253:
230:
210:
198:
195:
53:is idempotent.
15:
13:
10:
9:
6:
4:
3:
2:
4527:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4492:
4490:
4480:
4476:
4472:
4466:
4462:
4458:
4457:
4451:
4448:
4447:
4442:
4438:
4434:
4430:
4424:
4420:
4416:
4412:
4409:
4405:
4401:
4395:
4391:
4387:
4383:
4378:
4375:
4371:
4367:
4361:
4357:
4353:
4349:
4345:
4341:
4340:
4335:
4331:
4328:
4324:
4320:
4317:, Cambridge:
4316:
4309:
4304:
4301:
4297:
4293:
4287:
4283:
4278:
4276:
4272:
4268:
4267:
4263:
4251:
4247:
4240:
4234:
4231:
4227:
4221:
4218:
4214:
4210:
4205:
4202:
4198:
4194:
4191:
4186:
4181:
4177:
4176:
4171:
4165:
4162:
4158:
4154:
4150:
4147:
4141:
4138:
4134:
4118:
4098:
4088:
4085:
4070:
4067:
4064:
4061:
4055:
4049:
4046:
4037:
4031:
4025:
4005:
4002:
3996:
3990:
3987:
3978:
3972:
3966:
3956:
3953:
3936:
3933:
3930:
3927:
3921:
3918:
3915:
3909:
3903:
3900:
3897:
3891:
3888:
3885:
3879:
3876:
3873:
3867:
3864:
3861:
3858:
3855:
3849:
3846:
3843:
3837:
3834:
3831:
3825:
3822:
3819:
3813:
3807:
3804:
3801:
3776:
3773:
3770:
3749:
3729:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3669:
3649:
3639:
3636:
3630:
3627:
3621:
3614:
3611:
3605:
3599:
3594:
3593:
3584:
3581:
3577:
3575:
3571:
3567:
3563:
3560:tout élément
3559:
3555:
3549:
3548:
3540:
3537:
3533:
3531:
3527:
3523:
3519:
3513:
3511:9781461209010
3507:
3503:
3502:
3494:
3491:
3487:
3483:
3478:
3475:
3465:
3464:
3457:
3452:
3448:
3444:
3440:
3437:(1): 97–229.
3436:
3432:
3428:
3422:
3421:
3416:
3412:
3408:
3402:
3399:
3394:
3390:
3389:
3384:
3378:
3375:
3370:
3369:
3364:
3363:"idempotence"
3358:
3355:
3349:
3345:
3342:
3340:
3339:Pure function
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3278:Biordered set
3276:
3275:
3271:
3269:
3267:
3263:
3254:
3247:
3245:
3243:
3239:
3235:
3231:
3226:
3224:
3219:
3217:
3212:
3210:
3206:
3202:
3197:
3195:
3190:
3186:
3184:
3183:
3178:
3174:
3170:
3005:
3003:
2997:
2994:
2986:
2984:
2977:
2973:
2972:pure function
2969:
2965:
2962:
2957:
2953:
2949:
2945:
2944:
2943:
2941:
2937:
2931:
2927:
2923:
2919:
2911:
2909:
2907:
2887:
2881:
2878:
2872:
2866:
2841:
2835:
2810:
2765:
2762:
2759:
2737:
2734:
2731:
2707:
2704:
2701:
2692:
2686:
2680:
2672:
2656:
2653:
2647:
2641:
2631:
2629:
2624:
2619:
2615:
2594:
2591:
2588:
2584:
2572:
2569:
2556:
2551:
2548:
2545:
2541:
2533:
2532:
2531:
2515:
2512:
2509:
2505:
2484:
2463:
2460:
2457:
2436:
2416:
2396:
2384:
2380:
2376:
2375:endomorphisms
2372:
2369:
2365:
2361:
2358:
2354:
2350:
2346:
2343:
2339:
2335:
2331:
2328:
2324:
2321:
2302:
2283:
2280:
2276:
2272:
2268:
2265:
2261:
2258:
2255:
2239:
2216:
2210:
2207:
2204:
2195:
2189:
2186:
2180:
2177:
2155:
2152:
2149:
2146:
2143:
2134:
2130:
2129:
2128:
2114:
2106:
2090:
2087:
2084:
2061:
2055:
2034:
2031:
2028:
2005:
1999:
1996:
1987:
1981:
1975:
1953:
1950:
1947:
1944:
1941:
1919:
1913:
1910:
1907:
1886:
1879:
1875:
1859:
1836:
1833:
1828:
1824:
1808:
1803:
1799:
1795:
1791:
1787:
1784:
1780:
1777:
1773:
1770:
1766:
1740:
1736:
1717:
1714:
1711:
1705:
1702:
1682:
1679:
1676:
1673:
1670:
1645:
1642:
1639:
1633:
1630:
1610:
1607:
1604:
1601:
1598:
1577:
1557:
1537:
1530:
1514:
1507:
1503:
1484:
1481:
1475:
1472:
1469:
1440:
1437:
1431:
1428:
1425:
1411:
1392:
1379:
1376:
1356:
1353:
1350:
1347:
1344:
1319:
1306:
1303:
1283:
1280:
1277:
1274:
1271:
1250:
1230:
1210:
1203:
1187:
1180:
1164:
1141:
1124:
1105:
1102:
1096:
1060:
1057:
1051:
1030:
1015:
1007:
991:
988:
985:
964:
961:
958:
955:
952:
949:
946:
924:
921:
918:
915:
912:
891:
871:
851:
828:
825:
822:
812:
808:
792:
789:
786:
783:
780:
758:
755:
752:
749:
746:
725:
718:
702:
695:
676:
673:
670:
660:
656:
637:
629:
625:
606:
603:
588:
584:
568:
565:
562:
559:
556:
534:
531:
528:
525:
522:
501:
481:
473:
469:
450:
447:
432:
428:
427:
423:
405:
402:
399:
379:
376:
373:
370:
367:
359:
358:
357:
355:
339:
332:
311:
308:
305:
302:
299:
291:
290:
289:
275:
267:
251:
244:
228:
208:
196:
194:
192:
191:
186:
185:
180:
175:
173:
169:
165:
161:
157:
153:
149:
145:
139:
115:
107:
65:
59:
52:
48:
44:
40:
36:
32:
26:
22:
4455:
4445:
4418:
4381:
4355:
4337:
4334:"Idempotent"
4314:
4281:
4250:the original
4245:
4233:
4220:
4204:
4196:
4174:
4164:
4155:. See also
4140:
4087:
3955:
3638:
3629:
3619:
3613:
3591:
3583:
3573:
3569:
3565:
3561:
3557:
3553:
3551:
3546:
3539:
3529:
3525:
3521:
3517:
3515:
3500:
3493:
3477:
3462:
3458:Reprinted:
3434:
3430:
3414:
3410:
3401:
3386:
3383:"idempotent"
3377:
3366:
3357:
3259:
3227:
3220:
3213:
3198:
3191:
3187:
3181:
3177:HTTP methods
3166:
3001:
2998:
2990:
2981:
2956:side effects
2939:
2933:
2632:
2617:
2611:
2388:
2379:vector space
2338:affine space
1812:
1776:Boolean ring
978:and finally
353:
330:
328:
265:
200:
188:
176:
57:
56:
50:
46:
38:
34:
4440:p. 443
4415:Lang, Serge
3516:An element
3455:See p. 104.
3182:nullipotent
2940:idempotence
2938:, the term
2926:Stable sort
2497:, and then
2389:If the set
2383:projections
2368:Kleene plus
2364:Kleene star
2334:convex hull
2105:fixed point
1794:determinant
1157:of the set
201:An element
148:mathematics
58:Idempotence
4489:Categories
4437:0848.13001
4327:0898.16032
4271:idempotent
3530:idempotent
3528:is called
3484:, p.
3423:Printed:
3415:idempotent
3350:References
3205:page fault
2952:subroutine
2916:See also:
2169:, that is
1933:such that
1739:GCD domain
904:such that
354:idempotent
266:idempotent
197:Definition
160:projectors
144:operations
4344:EMS Press
4273:" at the
4131:is not a
4068:≠
3934:∘
3919:∘
3910:∘
3901:∘
3886:∘
3877:∘
3868:∘
3856:∘
3847:∘
3838:∘
3823:∘
3814:∘
3805:∘
3774:∘
3706:∘
3694:∘
3411:nilpotent
3334:Nilpotent
2888:⋅
2882:−
2879:∘
2873:⋅
2867:−
2842:⋅
2836:−
2814:¬
2811:∘
2808:¬
2787:¬
2763:∘
2735:∘
2592:−
2542:∑
2513:−
2461:−
2340:over the
2211:
2190:
2181:
2147:∘
2088:∈
2032:∈
1945:∘
1917:→
1911::
1887:∘
1837:∘
1706:∈
1674:∧
1634:∈
1602:∨
1578:∧
1558:∨
1538:∧
1515:∨
1485:∧
1441:∨
1380:∈
1348:∩
1307:∈
1275:∪
1251:∩
1231:∪
1211:∩
1188:∪
1179:set union
1123:power set
1106:∩
1061:∪
962:⋅
950:⋅
916:⋅
829:⋅
784:⋅
750:⋅
677:⋅
652:0 + 0 = 0
560:×
526:×
451:×
403:∈
371:⋅
340:⋅
303:⋅
276:⋅
252:⋅
221:of a set
4417:(1993),
4149:Archived
3790:, since
3568:tel que
3393:Archived
3272:See also
3262:elevator
3144:sequence
3135:sequence
3093:sequence
3042:"%d
2993:database
2381:are its
2353:interior
2257:constant
2232:for all
2021:for all
1695:for all
1623:for all
1369:for all
1296:for all
628:addition
424:Examples
392:for all
4479:1896125
4419:Algebra
4408:1838439
4374:2106764
4346:, 2001
4300:1150975
3451:2369153
3244:, etc.
3167:In the
3114:inspect
3102:inspect
3027:inspect
2626:in the
2623:A000248
2349:closure
2275:ceiling
1876:) with
1500:of the
1121:of the
938:, then
630:, only
622:of the
585:In the
474:, only
466:of the
429:In the
190:potence
4477:
4467:
4435:
4425:
4406:
4396:
4372:
4362:
4325:
4298:
4288:
4144:IETF,
4018:, but
3600:
3508:
3449:
3173:safety
3153:return
3108:change
3066:change
3048:"
3036:printf
2928:, and
2673:3 and
1796:of an
1792:, the
1662:, and
1336:, and
715:or an
587:monoid
431:monoid
268:under
166:) and
4449:1870.
4311:(PDF)
4253:(PDF)
4242:(PDF)
3959:e.g.
3552:Soit
3467:(PDF)
3447:JSTOR
3199:In a
2954:with
2377:of a
2342:reals
2318:of a
2271:floor
2103:is a
1788:In a
1781:In a
1774:In a
1737:In a
1504:with
1177:with
811:group
809:In a
692:, an
659:magma
657:In a
626:with
470:with
4465:ISBN
4423:ISBN
4394:ISBN
4360:ISBN
4286:ISBN
4193:7231
4111:and
3742:and
3662:and
3598:ISBN
3506:ISBN
3126:main
3090:void
3063:void
3024:void
2970:, a
2950:, a
2628:OEIS
2409:has
2366:and
2362:the
2351:and
2347:the
2332:the
2325:the
2277:and
2269:the
2262:the
2131:the
1767:and
1570:and
1527:and
1456:and
1243:and
1200:and
1076:and
772:and
548:and
494:and
329:The
184:idem
162:and
150:and
23:and
4461:127
4433:Zbl
4386:doi
4323:Zbl
4190:RFC
4180:doi
3642:If
3564:de
3486:127
3439:doi
3232:,
3221:In
3192:In
3147:();
3138:();
3123:int
3117:();
3111:();
3105:();
3009:int
3000:on—
2966:in
2946:in
2934:In
2696:max
2671:mod
2630:).
2208:abs
2187:abs
2178:abs
2156:abs
2150:abs
2144:abs
2107:of
1769:LCM
1765:GCD
1008:of
356:if
288:if
174:).
146:in
51:Off
39:Off
4491::
4475:MR
4473:,
4463:,
4431:,
4404:MR
4402:,
4392:,
4370:MR
4368:,
4342:,
4336:,
4296:MR
4294:,
4244:.
4211:.
4195:.
4188:.
4172:.
3572:=
3524:=
3522:ss
3514:.
3445:.
3433:.
3429:.
3391:.
3385:.
3365:.
3129:()
3096:()
3069:()
3057:);
3045:\n
3030:()
2924:,
2920:,
2273:,
187:+
138:-/
126:aɪ
116::
114:US
110:,
100:ən
94:oʊ
66::
64:UK
47:On
35:On
4388::
4269:"
4215:.
4182::
4159:.
4119:g
4099:f
4071:1
4065:2
4062:=
4059:)
4056:5
4053:(
4050:f
4047:=
4044:)
4041:)
4038:1
4035:(
4032:g
4029:(
4026:f
4006:1
4003:=
4000:)
3997:7
3994:(
3991:f
3988:=
3985:)
3982:)
3979:7
3976:(
3973:g
3970:(
3967:f
3937:g
3931:f
3928:=
3925:)
3922:g
3916:g
3913:(
3907:)
3904:f
3898:f
3895:(
3892:=
3889:g
3883:)
3880:g
3874:f
3871:(
3865:f
3862:=
3859:g
3853:)
3850:f
3844:g
3841:(
3835:f
3832:=
3829:)
3826:g
3820:f
3817:(
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