42:
3015:. 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.
3264:
3196:). 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.
3200:
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.
3190:. 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
3279:. 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.
3006:
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
3199:
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
2993:
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.
3958:
2969:
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
3430:
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."
3007:
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.
2241:
2177:
3010:
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
3805:
3239:
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,
2618:
3222:
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.
1130:
1085:
2912:
1417:
1344:
475:
4092:
192:
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
3730:
1509:
1166:
986:
2835:
2327:
631:
1465:
4027:
2732:
1941:
1861:
2030:
1772:
1742:
1670:
1704:
590:
556:
1975:
946:
853:
814:
780:
701:
401:
333:
2866:
1632:
1378:
1305:
3798:
2787:
2759:
2539:
1599:
1559:
2678:
2112:
2056:
1908:
427:
361:
297:
273:
2808:
1579:
1536:
1272:
1252:
1232:
1209:
2086:
2485:
1013:
165:
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
4140:
4120:
3771:
3751:
3691:
3671:
2506:
2458:
2438:
2418:
2261:
2136:
1881:
1186:
1037:
913:
893:
873:
747:
724:
659:
523:
503:
250:
230:
3236:(SOA), a multiple-step orchestration process composed entirely of idempotent steps can be replayed without side-effects if any part of that process fails.
2184:
2150:
4510:
3644:
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.
2638:
4505:
4208:
It knows that repeating the request will have the same intended effect, even if the original request succeeded, though the response might differ.
3207:, 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.
4525:
4479:
4437:
4408:
4374:
4300:
3612:
4285:
3472:
3229:
is expected to be idempotent. In other words, if the output is already "pretty", there should be nothing to do for the pretty-printer.
4268:
For example, this design specification includes detailed algorithm for when elevator cars will respond to subsequent calls for service
2644:
Neither the property of being idempotent nor that of being not is preserved under function composition. As an example for the former,
3520:
3403:
3953:{\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}
60:
button (green) is an idempotent operation, since it has the same effect whether done once or multiple times. Likewise, pressing
4520:
4236:
2966:
4354:
4249:
3233:
2547:
3416:
Original manuscript of 1870 lecture before
National Academy of Sciences (Washington, DC, USA): Peirce, Benjamin (1870)
3211:
4349:
4167:
3187:
3183:
3179:
2393:
2285:
2281:
3420:
From pages 16-17: "When an expression which is raised to the square or any higher power vanishes, it may be called
4515:
4329:
4223:
170:
3378:
3298:
2337:
2115:
1212:
4471:
4465:
4159:
3496:
3354:
3329:
3324:
2940:
2928:
1090:
1045:
182:
31:
2873:
41:
4369:, Mathematics and its Applications, vol. 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380,
3204:
1775:
1383:
1310:
446:
154:
4032:
2978:
2958:
2932:
1884:
178:
4395:, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385,
1779:
3697:
3263:
1470:
1138:
953:
2814:
2298:
602:
4200:
3303:
2541:
is the number of different idempotent functions. Hence, taking into account all possible partitions,
1888:
1426:
3973:
2687:
2631:= 0, 1, 2, 3, 4, 5, 6, 7, 8, ... starts with 1, 1, 3, 10, 41, 196, 1057, 6322, 41393, ... (sequence
1914:
1827:
4362:
4143:
3437:
3318:
3308:
2363:
1539:
1516:
821:
1982:
1755:
1709:
1637:
3457:
2681:
2359:
1677:
1189:
563:
529:
4344:
1948:
919:
826:
787:
753:
674:
374:
306:
2842:
1605:
1351:
1278:
4475:
4433:
4404:
4370:
4296:
4180:
3777:
3608:
3516:
3424:; but when raised to a square or higher power it gives itself as the result, it may be called
3334:
3313:
2916:
2766:
2738:
2367:
2274:
2267:
1808:
1793:
727:
35:
4455:
3417:
2511:
1584:
1544:
4443:
4396:
4333:
4190:
3449:
3339:
3293:
3276:
3219:
2946:
2648:
2624:
2091:
2035:
1893:
704:
406:
346:
282:
258:
174:
166:
162:
130:
124:
80:
53:
4489:
4418:
4384:
4318:
4310:
2793:
1564:
1521:
1257:
1237:
1217:
1194:
4485:
4447:
4414:
4380:
4337:
4306:
4219:
4163:
3398:
3252:
2289:
2062:
1812:
1016:
669:
253:
189:
74:
2464:
992:
3191:
4125:
4105:
3756:
3736:
3676:
3656:
3601:
3393:
3248:
2491:
2443:
2423:
2403:
2330:
2246:
2143:
2121:
1866:
1512:
1171:
1022:
898:
878:
858:
732:
709:
644:
634:
508:
488:
482:
478:
235:
215:
4295:(2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412,
4499:
4237:"Compiler construction of idempotent regions and applications in architecture design"
3349:
3288:
2982:
2236:{\displaystyle \operatorname {abs} (\operatorname {abs} (x))=\operatorname {abs} (x)}
3271:
Applied examples that many people could encounter in their day-to-day lives include
4156:
2389:
2385:
2348:
2172:{\displaystyle \operatorname {abs} \circ \operatorname {abs} =\operatorname {abs} }
1786:
4250:"Geared Traction Passenger Elevator Specification Guide Information/Instructions"
3556:
3510:
4203:
4184:
3226:
2936:
2378:
2374:
2352:
2344:
1804:
1800:
158:
4470:, Algebras and Applications, vol. 1, Kluwer Academic Publishers, pp.
4260:
4425:
4400:
3215:
2962:
2953:
may have a different meaning depending on the context in which it is applied:
1749:
3344:
1133:
17:
200:
194:
3272:
3003:
1811:
is either 0 or 1. If the determinant is 1, the matrix necessarily is the
638:
3461:
3373:
3321: – a generalization of idempotence to binary relations
4195:
3240:
2985:
is idempotent if it is idempotent in the mathematical sense given in
2623:
is the total number of possible idempotent functions on the set. The
597:
441:
3453:
2627:
of the number of idempotent functions as given by the sum above for
3633:. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc.
2790:
happens to be. As an example for the latter, the negation function
3262:
3244:
40:
4326:
Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994
3267:
A typical crosswalk button is an example of an idempotent system
2381:
functions of the power set of a monoid to itself are idempotent;
4281:
3251:, installing an application and all of its dependencies with a
2340:
function from the power set of a group to itself is idempotent;
110:
4186:
Hypertext
Transfer Protocol (HTTP/1.1): Semantics and Content
4157:
Hypertext
Transfer Protocol (HTTP/1.1): Semantics and Content
3515:. Berlin: Springer Science & Business Media. p. 22.
1395:
1322:
1144:
1099:
1054:
2633:
142:
136:
104:
92:
3567:
un magma, noté multiplicativement. On nomme idempotent de
3002:
A function looking up a customer's name and address in a
86:
3512:
Linear
Algebra: An Introduction to Abstract Mathematics
3480:. New York, New York, USA: D. Van Nostrand. p. 8.
3013:
idempotence is not closed under sequential composition
4128:
4108:
4035:
3976:
3808:
3780:
3759:
3739:
3700:
3679:
3659:
2876:
2845:
2817:
2796:
2769:
2741:
2690:
2651:
2550:
2514:
2494:
2467:
2446:
2426:
2406:
2301:
2249:
2187:
2153:
2124:
2094:
2065:
2038:
1985:
1951:
1917:
1896:
1869:
1830:
1758:
1712:
1680:
1640:
1608:
1587:
1567:
1547:
1524:
1473:
1429:
1386:
1354:
1313:
1281:
1260:
1240:
1220:
1197:
1174:
1141:
1093:
1048:
1025:
995:
956:
922:
901:
881:
861:
829:
790:
756:
735:
712:
677:
647:
605:
566:
532:
511:
491:
449:
409:
377:
349:
309:
285:
261:
238:
218:
113:
83:
4464:
Polcino Milies, César; Sehgal, Sudarshan K. (2002),
139:
107:
101:
89:
2613:{\displaystyle \sum _{k=0}^{n}{n \choose k}k^{n-k}}
145:
133:
98:
95:
4432:(Third ed.), Reading, Mass.: Addison-Wesley,
4134:
4114:
4086:
4021:
3952:
3792:
3765:
3745:
3724:
3685:
3665:
3600:
2906:
2860:
2829:
2802:
2781:
2753:
2726:
2672:
2612:
2533:
2500:
2479:
2452:
2432:
2412:
2321:
2255:
2235:
2171:
2130:
2106:
2080:
2050:
2024:
1969:
1935:
1902:
1875:
1855:
1766:
1736:
1698:
1664:
1626:
1593:
1573:
1553:
1530:
1503:
1459:
1411:
1372:
1338:
1299:
1266:
1246:
1226:
1203:
1180:
1160:
1124:
1079:
1031:
1007:
980:
940:
907:
887:
867:
847:
808:
774:
741:
718:
695:
653:
625:
584:
550:
517:
497:
469:
421:
395:
355:
327:
291:
267:
244:
224:
188:The term was introduced by American mathematician
2915:is. In both cases, the composition is simply the
2588:
2575:
2706:
3218:are idempotent. So if a page fault occurs, the
4365:; Gubareni, Nadiya; Kirichenko, V. V. (2004),
3382:(3rd ed.). Oxford University Press. 2010.
3492:
2810:on the Boolean domain is not idempotent, but
181:(in which it is connected to the property of
8:
1731:
1719:
1659:
1647:
1489:
1477:
1445:
1433:
3214:, instructions that might possibly cause a
875:is the only idempotent element. Indeed, if
3561:(in French). Paris: Vuibert. p. 180.
4194:
4127:
4107:
4034:
3975:
3961:, using the associativity of composition.
3807:
3779:
3758:
3738:
3699:
3678:
3658:
2875:
2844:
2816:
2795:
2768:
2740:
2689:
2650:
2598:
2587:
2574:
2572:
2566:
2555:
2549:
2519:
2513:
2493:
2466:
2445:
2425:
2405:
2302:
2300:
2248:
2186:
2152:
2123:
2093:
2064:
2037:
1984:
1950:
1916:
1895:
1868:
1838:
1829:
1760:
1759:
1757:
1711:
1679:
1639:
1607:
1586:
1566:
1546:
1523:
1472:
1428:
1394:
1393:
1385:
1353:
1321:
1320:
1312:
1280:
1259:
1239:
1219:
1196:
1173:
1143:
1142:
1140:
1125:{\displaystyle ({\mathcal {P}}(E),\cap )}
1098:
1097:
1092:
1080:{\displaystyle ({\mathcal {P}}(E),\cup )}
1053:
1052:
1047:
1024:
994:
955:
921:
900:
880:
860:
828:
789:
755:
734:
711:
676:
646:
610:
609:
604:
565:
531:
510:
490:
454:
453:
448:
408:
376:
348:
308:
284:
260:
237:
217:
2907:{\displaystyle -(\cdot )\circ -(\cdot )}
2869:of real numbers is not idempotent, but
1910:, idempotent elements are the functions
4257:NC Department Of Labor, Elevator Bureau
3365:
3186:are the major attributes that separate
749:, if it exists, is idempotent. Indeed,
4393:A first course in noncommutative rings
1412:{\displaystyle x\in {\mathcal {P}}(E)}
1339:{\displaystyle x\in {\mathcal {P}}(E)}
470:{\displaystyle (\mathbb {N} ,\times )}
7:
4286:Free On-line Dictionary of Computing
4087:{\displaystyle f(g(1))=f(5)=2\neq 1}
4367:Algebras, rings and modules. vol. 1
3693:commute under composition (i.e. if
2440:elements, we can partition it into
2824:
2818:
2797:
2579:
2347:function from the power set of an
2306:
2303:
1015:by multiplying on the left by the
25:
4102:also showing that commutation of
3725:{\displaystyle f\circ g=g\circ f}
1504:{\displaystyle (\{0,1\},\wedge )}
1161:{\displaystyle {\mathcal {P}}(E)}
981:{\displaystyle x\cdot x=x\cdot e}
169:(in particular, in the theory of
30:For the concepts in algebra, see
4511:Algebraic properties of elements
4324:, in Gunawardena, Jeremy (ed.),
4319:"An introduction to idempotency"
3406:from the original on 2016-10-19.
2830:{\displaystyle \neg \circ \neg }
2366:functions of the power set of a
2322:{\displaystyle \mathrm {Re} (z)}
626:{\displaystyle (\mathbb {N} ,+)}
129:
79:
4506:Properties of binary operations
3442:American Journal of Mathematics
2986:
2971:
1789:, multiplication is idempotent.
1460:{\displaystyle (\{0,1\},\vee )}
4467:An Introduction to Group Rings
4069:
4063:
4054:
4051:
4045:
4039:
4022:{\displaystyle f(g(7))=f(7)=1}
4010:
4004:
3995:
3992:
3986:
3980:
3935:
3923:
3917:
3905:
3893:
3881:
3863:
3851:
3839:
3827:
3821:
3809:
2901:
2895:
2886:
2880:
2855:
2849:
2838:is. Similarly, unary negation
2727:{\displaystyle g(x)=\max(x,5)}
2721:
2709:
2700:
2694:
2661:
2655:
2316:
2310:
2230:
2224:
2212:
2209:
2203:
2194:
2075:
2069:
2019:
2013:
2004:
2001:
1995:
1989:
1936:{\displaystyle f\colon E\to E}
1927:
1856:{\displaystyle (E^{E},\circ )}
1850:
1831:
1498:
1474:
1454:
1430:
1406:
1400:
1333:
1327:
1155:
1149:
1119:
1110:
1104:
1094:
1074:
1065:
1059:
1049:
842:
830:
690:
678:
620:
606:
464:
450:
1:
3558:Polynômes et algèbre linéaire
3234:service-oriented architecture
153:) is the property of certain
4526:Theoretical computer science
4317:Gunawardena, Jeremy (1998),
4146:for idempotency preservation
3438:"Linear associative algebra"
3418:"Linear associative algebra"
3161:// prints "5\n5\n"
3152:// prints "3\n5\n"
2025:{\displaystyle f(f(x))=f(x)}
1863:of the functions from a set
1767:{\displaystyle \mathbb {Z} }
1737:{\displaystyle x\in \{0,1\}}
1665:{\displaystyle x\in \{0,1\}}
56:control panel. Pressing the
4350:Encyclopedia of Mathematics
4168:HyperText Transfer Protocol
3733:) then idempotency of both
3180:Hypertext Transfer Protocol
2059:(in other words, the image
1699:{\displaystyle x\wedge x=x}
585:{\displaystyle 1\times 1=1}
551:{\displaystyle 0\times 0=0}
4542:
4457:Linear Associative Algebra
4330:Cambridge University Press
3474:Linear Associative Algebra
3225:When reformatting output,
2926:
1970:{\displaystyle f\circ f=f}
1801:ring of quadratic matrices
941:{\displaystyle x\cdot x=x}
848:{\displaystyle (G,\cdot )}
809:{\displaystyle a\cdot a=a}
775:{\displaystyle e\cdot e=e}
696:{\displaystyle (M,\cdot )}
396:{\displaystyle x\cdot x=x}
328:{\displaystyle x\cdot x=x}
29:
4401:10.1007/978-1-4419-8616-0
4293:von Neumann regular rings
3629:Garrett Birkhoff (1967).
3493:Polcino & Sehgal 2002
3471:Peirce, Benjamin (1882).
3436:Peirce, Benjamin (1881).
3379:Oxford English Dictionary
3299:Fixed point (mathematics)
2998:Computer science examples
2861:{\displaystyle -(\cdot )}
2734:are both idempotent, but
2370:to itself are idempotent;
2292:functions are idempotent;
2270:functions are idempotent;
1796:, addition is idempotent.
1627:{\displaystyle x\vee x=x}
1373:{\displaystyle x\cap x=x}
1300:{\displaystyle x\cup x=x}
4291:Goodearl, K. R. (1991),
3793:{\displaystyle f\circ g}
3555:Doneddu, Alfred (1976).
3509:Valenza, Robert (2012).
3355:Referential transparency
3330:Involution (mathematics)
3325:Idempotent (ring theory)
3241:resuming a file transfer
3182:(HTTP), idempotence and
3017:
2941:Command query separation
2929:Referential transparency
2923:Computer science meaning
2782:{\displaystyle g\circ f}
2754:{\displaystyle f\circ g}
2460:chosen fixed points and
2355:to itself is idempotent;
1601:are idempotent. Indeed,
1274:are idempotent. Indeed,
525:are idempotent. Indeed,
183:referential transparency
32:Idempotent (ring theory)
3599:George Grätzer (2003).
3212:load–store architecture
3205:event stream processing
2919:, which is idempotent.
2534:{\displaystyle k^{n-k}}
2488:non-fixed points under
2295:the real part function
2146:is idempotent. Indeed,
1594:{\displaystyle \wedge }
1554:{\displaystyle \wedge }
855:, the identity element
661:is idempotent. Indeed,
4521:Mathematical relations
4259:. 2002. Archived from
4136:
4116:
4088:
4023:
3954:
3794:
3767:
3747:
3726:
3687:
3667:
3635:. Here: Sect.I.5, p.8.
3603:General Lattice Theory
3268:
2979:functional programming
2959:imperative programming
2933:Reentrant (subroutine)
2908:
2862:
2831:
2804:
2783:
2755:
2728:
2674:
2673:{\displaystyle f(x)=x}
2614:
2571:
2535:
2502:
2481:
2454:
2434:
2414:
2323:
2257:
2237:
2173:
2132:
2108:
2107:{\displaystyle x\in E}
2082:
2052:
2051:{\displaystyle x\in E}
2026:
1971:
1937:
1904:
1903:{\displaystyle \circ }
1877:
1857:
1768:
1738:
1700:
1666:
1628:
1595:
1575:
1555:
1532:
1505:
1461:
1413:
1374:
1340:
1301:
1268:
1248:
1228:
1205:
1182:
1162:
1126:
1081:
1033:
1009:
982:
942:
909:
889:
869:
849:
810:
776:
743:
720:
697:
655:
627:
586:
552:
519:
499:
471:
423:
422:{\displaystyle x\in S}
397:
357:
356:{\displaystyle \cdot }
329:
293:
292:{\displaystyle \cdot }
269:
268:{\displaystyle \cdot }
246:
226:
193:
179:functional programming
65:
27:Property of operations
4137:
4117:
4089:
4024:
3955:
3795:
3768:
3748:
3727:
3688:
3668:
3607:. Basel: Birkhäuser.
3531:of a magma such that
3266:
2909:
2863:
2832:
2805:
2803:{\displaystyle \neg }
2784:
2756:
2729:
2675:
2615:
2551:
2536:
2503:
2482:
2455:
2435:
2415:
2324:
2258:
2238:
2174:
2133:
2109:
2083:
2053:
2027:
1972:
1938:
1905:
1878:
1858:
1774:), the operations of
1769:
1739:
1701:
1667:
1629:
1596:
1576:
1574:{\displaystyle \vee }
1556:
1533:
1531:{\displaystyle \vee }
1506:
1462:
1414:
1375:
1341:
1302:
1269:
1267:{\displaystyle \cap }
1249:
1247:{\displaystyle \cup }
1229:
1227:{\displaystyle \cap }
1206:
1204:{\displaystyle \cup }
1183:
1163:
1127:
1082:
1034:
1010:
983:
943:
910:
890:
870:
850:
811:
777:
744:
721:
698:
656:
628:
587:
553:
520:
500:
472:
424:
398:
358:
330:
294:
270:
247:
227:
52:buttons of a train's
44:
4181:"Idempotent Methods"
4126:
4106:
4033:
3974:
3806:
3778:
3757:
3737:
3698:
3677:
3657:
3619:Here: Sect.1.2, p.5.
3304:Idempotent of a code
2874:
2843:
2815:
2794:
2767:
2739:
2688:
2649:
2548:
2512:
2492:
2465:
2444:
2424:
2404:
2299:
2247:
2185:
2151:
2122:
2092:
2081:{\displaystyle f(x)}
2063:
2036:
1983:
1978:, that is such that
1949:
1915:
1894:
1889:function composition
1867:
1828:
1820:Idempotent functions
1756:
1710:
1678:
1638:
1606:
1585:
1565:
1545:
1522:
1471:
1427:
1384:
1352:
1311:
1279:
1258:
1238:
1218:
1195:
1172:
1139:
1091:
1046:
1023:
993:
954:
920:
899:
879:
859:
827:
788:
754:
733:
710:
675:
645:
603:
564:
530:
509:
489:
447:
407:
375:
347:
307:
283:
259:
236:
216:
4391:Lam, T. Y. (2001),
4363:Hazewinkel, Michiel
4235:Marc A. de Kruijf.
4189:. sec. 4.2.2.
4144:necessary condition
3319:Idempotent relation
3309:Idempotent analysis
3245:synchronizing files
2480:{\displaystyle n-k}
1540:logical conjunction
1517:logical disjunction
1008:{\displaystyle x=e}
4454:Peirce, Benjamin.
4162:2014-06-08 at the
4132:
4112:
4084:
4019:
3950:
3790:
3763:
3743:
3722:
3683:
3663:
3269:
2904:
2858:
2827:
2800:
2779:
2751:
2724:
2670:
2610:
2531:
2498:
2477:
2450:
2430:
2410:
2338:subgroup generated
2319:
2253:
2233:
2169:
2128:
2104:
2078:
2048:
2022:
1967:
1933:
1900:
1885:set exponentiation
1873:
1853:
1764:
1734:
1696:
1662:
1624:
1591:
1571:
1551:
1528:
1501:
1457:
1409:
1370:
1336:
1297:
1264:
1244:
1224:
1201:
1178:
1158:
1122:
1077:
1029:
1005:
978:
938:
905:
885:
865:
845:
806:
772:
739:
716:
693:
651:
623:
582:
548:
515:
495:
467:
419:
393:
353:
325:
289:
265:
242:
222:
66:
4516:Closure operators
4481:978-1-4020-0238-0
4439:978-0-201-55540-0
4410:978-0-387-95183-6
4376:978-1-4020-2690-4
4332:, pp. 1–49,
4302:978-0-89464-632-4
4135:{\displaystyle g}
4115:{\displaystyle f}
3766:{\displaystyle g}
3746:{\displaystyle f}
3686:{\displaystyle g}
3666:{\displaystyle f}
3614:978-3-7643-6996-5
3335:Iterated function
3314:Idempotent matrix
3277:crosswalk buttons
3275:call buttons and
2917:identity function
2762:is not, although
2586:
2501:{\displaystyle f}
2453:{\displaystyle k}
2433:{\displaystyle n}
2413:{\displaystyle E}
2368:topological space
2275:identity function
2256:{\displaystyle x}
2131:{\displaystyle f}
1876:{\displaystyle E}
1809:idempotent matrix
1794:Tropical semiring
1752:(for instance in
1181:{\displaystyle E}
1032:{\displaystyle x}
908:{\displaystyle G}
895:is an element of
888:{\displaystyle x}
868:{\displaystyle e}
742:{\displaystyle a}
728:absorbing element
719:{\displaystyle e}
654:{\displaystyle 0}
518:{\displaystyle 1}
498:{\displaystyle 0}
245:{\displaystyle S}
225:{\displaystyle x}
175:closure operators
36:Idempotent matrix
16:(Redirected from
4533:
4492:
4450:
4421:
4387:
4358:
4340:
4323:
4313:
4269:
4267:
4265:
4254:
4246:
4240:
4233:
4227:
4217:
4211:
4210:
4198:
4196:10.17487/RFC7231
4177:
4171:
4153:
4147:
4141:
4139:
4138:
4133:
4121:
4119:
4118:
4113:
4100:
4094:
4093:
4091:
4090:
4085:
4028:
4026:
4025:
4020:
3968:
3962:
3960:
3959:
3957:
3956:
3951:
3800:
3799:
3797:
3796:
3791:
3773:implies that of
3772:
3770:
3769:
3764:
3752:
3750:
3749:
3744:
3732:
3731:
3729:
3728:
3723:
3692:
3690:
3689:
3684:
3672:
3670:
3669:
3664:
3651:
3645:
3642:
3636:
3634:
3626:
3620:
3618:
3606:
3596:
3590:
3589:
3552:
3546:
3545:
3506:
3500:
3490:
3484:
3481:
3479:
3465:
3414:
3408:
3407:
3390:
3384:
3383:
3370:
3340:List of matrices
3294:Closure operator
3259:Applied examples
3220:operating system
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
2947:computer science
2914:
2913:
2911:
2910:
2905:
2868:
2867:
2865:
2864:
2859:
2837:
2836:
2834:
2833:
2828:
2809:
2807:
2806:
2801:
2789:
2788:
2786:
2785:
2780:
2761:
2760:
2758:
2757:
2752:
2733:
2731:
2730:
2725:
2680:
2679:
2677:
2676:
2671:
2636:
2625:integer sequence
2619:
2617:
2616:
2611:
2609:
2608:
2593:
2592:
2591:
2578:
2570:
2565:
2540:
2538:
2537:
2532:
2530:
2529:
2507:
2505:
2504:
2499:
2487:
2486:
2484:
2483:
2478:
2459:
2457:
2456:
2451:
2439:
2437:
2436:
2431:
2419:
2417:
2416:
2411:
2333:, is idempotent.
2328:
2326:
2325:
2320:
2309:
2263:
2262:
2260:
2259:
2254:
2242:
2240:
2239:
2234:
2179:
2178:
2176:
2175:
2170:
2138:). For example:
2137:
2135:
2134:
2129:
2113:
2111:
2110:
2105:
2088:of each element
2087:
2085:
2084:
2079:
2058:
2057:
2055:
2054:
2049:
2031:
2029:
2028:
2023:
1977:
1976:
1974:
1973:
1968:
1943:
1942:
1940:
1939:
1934:
1909:
1907:
1906:
1901:
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1843:
1842:
1773:
1771:
1770:
1765:
1763:
1744:
1743:
1741:
1740:
1735:
1705:
1703:
1702:
1697:
1672:
1671:
1669:
1668:
1663:
1633:
1631:
1630:
1625:
1600:
1598:
1597:
1592:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1537:
1535:
1534:
1529:
1510:
1508:
1507:
1502:
1466:
1464:
1463:
1458:
1419:
1418:
1416:
1415:
1410:
1399:
1398:
1379:
1377:
1376:
1371:
1346:
1345:
1343:
1342:
1337:
1326:
1325:
1306:
1304:
1303:
1298:
1273:
1271:
1270:
1265:
1253:
1251:
1250:
1245:
1233:
1231:
1230:
1225:
1213:set intersection
1210:
1208:
1207:
1202:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1148:
1147:
1131:
1129:
1128:
1123:
1103:
1102:
1086:
1084:
1083:
1078:
1058:
1057:
1038:
1036:
1035:
1030:
1014:
1012:
1011:
1006:
988:
987:
985:
984:
979:
948:
947:
945:
944:
939:
914:
912:
911:
906:
894:
892:
891:
886:
874:
872:
871:
866:
854:
852:
851:
846:
816:
815:
813:
812:
807:
782:
781:
779:
778:
773:
748:
746:
745:
740:
725:
723:
722:
717:
705:identity element
702:
700:
699:
694:
664:
660:
658:
657:
652:
632:
630:
629:
624:
613:
592:
591:
589:
588:
583:
558:
557:
555:
554:
549:
524:
522:
521:
516:
504:
502:
501:
496:
476:
474:
473:
468:
457:
429:
428:
426:
425:
420:
402:
400:
399:
394:
362:
360:
359:
354:
342:binary operation
335:
334:
332:
331:
326:
298:
296:
295:
290:
274:
272:
271:
266:
252:equipped with a
251:
249:
248:
243:
231:
229:
228:
223:
204:(same + power).
167:abstract algebra
163:computer science
152:
151:
148:
147:
144:
141:
138:
135:
128:
120:
119:
116:
115:
112:
109:
106:
103:
100:
97:
94:
91:
88:
85:
78:
54:destination sign
21:
4541:
4540:
4536:
4535:
4534:
4532:
4531:
4530:
4496:
4495:
4482:
4463:
4440:
4424:
4411:
4390:
4377:
4361:
4343:
4321:
4316:
4303:
4290:
4277:
4275:Further reading
4272:
4263:
4252:
4248:
4247:
4243:
4234:
4230:
4224:"Demand Paging"
4220:John Ousterhout
4218:
4214:
4179:
4178:
4174:
4164:Wayback Machine
4154:
4150:
4124:
4123:
4104:
4103:
4101:
4097:
4031:
4030:
3972:
3971:
3969:
3965:
3804:
3803:
3802:
3776:
3775:
3774:
3755:
3754:
3735:
3734:
3696:
3695:
3694:
3675:
3674:
3655:
3654:
3652:
3648:
3643:
3639:
3628:
3627:
3623:
3615:
3598:
3597:
3593:
3554:
3553:
3549:
3523:
3508:
3507:
3503:
3491:
3487:
3477:
3470:
3454:10.2307/2369153
3435:
3429:
3415:
3411:
3399:Merriam-Webster
3392:
3391:
3387:
3372:
3371:
3367:
3363:
3285:
3261:
3253:package manager
3227:pretty-printing
3176:
3175:
3172:
3169:
3166:
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:
3000:
2943:
2925:
2872:
2871:
2870:
2841:
2840:
2839:
2813:
2812:
2811:
2792:
2791:
2765:
2764:
2763:
2737:
2736:
2735:
2686:
2685:
2647:
2646:
2645:
2632:
2594:
2573:
2546:
2545:
2515:
2510:
2509:
2490:
2489:
2463:
2462:
2461:
2442:
2441:
2422:
2421:
2402:
2401:
2384:the idempotent
2297:
2296:
2290:fractional part
2245:
2244:
2183:
2182:
2181:
2149:
2148:
2147:
2120:
2119:
2090:
2089:
2061:
2060:
2034:
2033:
1981:
1980:
1979:
1947:
1946:
1945:
1913:
1912:
1911:
1892:
1891:
1883:to itself (see
1865:
1864:
1834:
1826:
1825:
1822:
1813:identity matrix
1782:are idempotent.
1754:
1753:
1708:
1707:
1676:
1675:
1674:
1636:
1635:
1604:
1603:
1602:
1583:
1582:
1563:
1562:
1543:
1542:
1520:
1519:
1469:
1468:
1425:
1424:
1423:In the monoids
1382:
1381:
1350:
1349:
1348:
1309:
1308:
1277:
1276:
1275:
1256:
1255:
1236:
1235:
1216:
1215:
1193:
1192:
1170:
1169:
1137:
1136:
1089:
1088:
1044:
1043:
1042:In the monoids
1021:
1020:
1017:inverse element
991:
990:
952:
951:
950:
918:
917:
916:
897:
896:
877:
876:
857:
856:
825:
824:
786:
785:
784:
752:
751:
750:
731:
730:
708:
707:
673:
672:
662:
643:
642:
635:natural numbers
601:
600:
562:
561:
560:
528:
527:
526:
507:
506:
487:
486:
479:natural numbers
445:
444:
437:
405:
404:
373:
372:
371:
345:
344:
305:
304:
303:
281:
280:
257:
256:
254:binary operator
234:
233:
214:
213:
210:
190:Benjamin Peirce
132:
123:
122:
82:
73:
72:
39:
28:
23:
22:
15:
12:
11:
5:
4539:
4537:
4529:
4528:
4523:
4518:
4513:
4508:
4498:
4497:
4494:
4493:
4480:
4461:
4452:
4438:
4422:
4409:
4388:
4375:
4359:
4341:
4314:
4301:
4288:
4276:
4273:
4271:
4270:
4266:on 2011-05-23.
4241:
4239:. 2012. p. 10.
4228:
4212:
4172:
4148:
4131:
4111:
4095:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3963:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3789:
3786:
3783:
3762:
3742:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3682:
3662:
3646:
3637:
3631:Lattice Theory
3621:
3613:
3591:
3547:
3521:
3501:
3485:
3483:
3482:
3467:
3409:
3385:
3364:
3362:
3359:
3358:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3316:
3311:
3306:
3301:
3296:
3291:
3284:
3281:
3260:
3257:
3249:software build
3247:, creating a
3018:
2999:
2996:
2991:
2990:
2987:the definition
2975:
2972:the definition
2924:
2921:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2857:
2854:
2851:
2848:
2826:
2823:
2820:
2799:
2778:
2775:
2772:
2750:
2747:
2744:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2669:
2666:
2663:
2660:
2657:
2654:
2621:
2620:
2607:
2604:
2601:
2597:
2590:
2585:
2582:
2577:
2569:
2564:
2561:
2558:
2554:
2528:
2525:
2522:
2518:
2497:
2476:
2473:
2470:
2449:
2429:
2409:
2398:
2397:
2382:
2371:
2356:
2341:
2334:
2331:complex number
2318:
2315:
2312:
2308:
2305:
2293:
2278:
2277:is idempotent;
2271:
2265:
2252:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2168:
2165:
2162:
2159:
2156:
2144:absolute value
2127:
2103:
2100:
2097:
2077:
2074:
2071:
2068:
2047:
2044:
2041:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1966:
1963:
1960:
1957:
1954:
1932:
1929:
1926:
1923:
1920:
1899:
1872:
1852:
1849:
1846:
1841:
1837:
1833:
1824:In the monoid
1821:
1818:
1817:
1816:
1797:
1790:
1783:
1762:
1746:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1695:
1692:
1689:
1686:
1683:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1623:
1620:
1617:
1614:
1611:
1590:
1570:
1561:respectively,
1550:
1527:
1513:Boolean domain
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1421:
1408:
1405:
1402:
1397:
1392:
1389:
1369:
1366:
1363:
1360:
1357:
1335:
1332:
1329:
1324:
1319:
1316:
1296:
1293:
1290:
1287:
1284:
1263:
1243:
1234:respectively,
1223:
1200:
1177:
1157:
1154:
1151:
1146:
1121:
1118:
1115:
1112:
1109:
1106:
1101:
1096:
1076:
1073:
1070:
1067:
1064:
1061:
1056:
1051:
1040:
1028:
1004:
1001:
998:
977:
974:
971:
968:
965:
962:
959:
937:
934:
931:
928:
925:
904:
884:
864:
844:
841:
838:
835:
832:
818:
805:
802:
799:
796:
793:
771:
768:
765:
762:
759:
738:
715:
692:
689:
686:
683:
680:
666:
650:
622:
619:
616:
612:
608:
594:
581:
578:
575:
572:
569:
547:
544:
541:
538:
535:
514:
494:
483:multiplication
466:
463:
460:
456:
452:
436:
433:
432:
431:
418:
415:
412:
392:
389:
386:
383:
380:
363:is said to be
352:
338:
337:
324:
321:
318:
315:
312:
288:
275:is said to be
264:
241:
221:
209:
206:
64:is idempotent.
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4538:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4503:
4501:
4491:
4487:
4483:
4477:
4473:
4469:
4468:
4462:
4459:
4458:
4453:
4449:
4445:
4441:
4435:
4431:
4427:
4423:
4420:
4416:
4412:
4406:
4402:
4398:
4394:
4389:
4386:
4382:
4378:
4372:
4368:
4364:
4360:
4356:
4352:
4351:
4346:
4342:
4339:
4335:
4331:
4328:, Cambridge:
4327:
4320:
4315:
4312:
4308:
4304:
4298:
4294:
4289:
4287:
4283:
4279:
4278:
4274:
4262:
4258:
4251:
4245:
4242:
4238:
4232:
4229:
4225:
4221:
4216:
4213:
4209:
4205:
4202:
4197:
4192:
4188:
4187:
4182:
4176:
4173:
4169:
4165:
4161:
4158:
4152:
4149:
4145:
4129:
4109:
4099:
4096:
4081:
4078:
4075:
4072:
4066:
4060:
4057:
4048:
4042:
4036:
4016:
4013:
4007:
4001:
3998:
3989:
3983:
3977:
3967:
3964:
3947:
3944:
3941:
3938:
3932:
3929:
3926:
3920:
3914:
3911:
3908:
3902:
3899:
3896:
3890:
3887:
3884:
3878:
3875:
3872:
3869:
3866:
3860:
3857:
3854:
3848:
3845:
3842:
3836:
3833:
3830:
3824:
3818:
3815:
3812:
3787:
3784:
3781:
3760:
3740:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3680:
3660:
3650:
3647:
3641:
3638:
3632:
3625:
3622:
3616:
3610:
3605:
3604:
3595:
3592:
3588:
3586:
3582:
3578:
3574:
3571:tout élément
3570:
3566:
3560:
3559:
3551:
3548:
3544:
3542:
3538:
3534:
3530:
3524:
3522:9781461209010
3518:
3514:
3513:
3505:
3502:
3498:
3494:
3489:
3486:
3476:
3475:
3468:
3463:
3459:
3455:
3451:
3448:(1): 97–229.
3447:
3443:
3439:
3433:
3432:
3427:
3423:
3419:
3413:
3410:
3405:
3401:
3400:
3395:
3389:
3386:
3381:
3380:
3375:
3374:"idempotence"
3369:
3366:
3360:
3356:
3353:
3351:
3350:Pure function
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3289:Biordered set
3287:
3286:
3282:
3280:
3278:
3274:
3265:
3258:
3256:
3254:
3250:
3246:
3242:
3237:
3235:
3230:
3228:
3223:
3221:
3217:
3213:
3208:
3206:
3201:
3197:
3195:
3194:
3189:
3185:
3181:
3016:
3014:
3008:
3005:
2997:
2995:
2988:
2984:
2983:pure function
2980:
2976:
2973:
2968:
2964:
2960:
2956:
2955:
2954:
2952:
2948:
2942:
2938:
2934:
2930:
2922:
2920:
2918:
2898:
2892:
2889:
2883:
2877:
2852:
2846:
2821:
2776:
2773:
2770:
2748:
2745:
2742:
2718:
2715:
2712:
2703:
2697:
2691:
2683:
2667:
2664:
2658:
2652:
2642:
2640:
2635:
2630:
2626:
2605:
2602:
2599:
2595:
2583:
2580:
2567:
2562:
2559:
2556:
2552:
2544:
2543:
2542:
2526:
2523:
2520:
2516:
2495:
2474:
2471:
2468:
2447:
2427:
2407:
2395:
2391:
2387:
2386:endomorphisms
2383:
2380:
2376:
2372:
2369:
2365:
2361:
2357:
2354:
2350:
2346:
2342:
2339:
2335:
2332:
2313:
2294:
2291:
2287:
2283:
2279:
2276:
2272:
2269:
2266:
2250:
2227:
2221:
2218:
2215:
2206:
2200:
2197:
2191:
2188:
2166:
2163:
2160:
2157:
2154:
2145:
2141:
2140:
2139:
2125:
2117:
2101:
2098:
2095:
2072:
2066:
2045:
2042:
2039:
2016:
2010:
2007:
1998:
1992:
1986:
1964:
1961:
1958:
1955:
1952:
1930:
1924:
1921:
1918:
1897:
1890:
1886:
1870:
1847:
1844:
1839:
1835:
1819:
1814:
1810:
1806:
1802:
1798:
1795:
1791:
1788:
1784:
1781:
1777:
1751:
1747:
1728:
1725:
1722:
1716:
1713:
1693:
1690:
1687:
1684:
1681:
1656:
1653:
1650:
1644:
1641:
1621:
1618:
1615:
1612:
1609:
1588:
1568:
1548:
1541:
1525:
1518:
1514:
1495:
1492:
1486:
1483:
1480:
1451:
1448:
1442:
1439:
1436:
1422:
1403:
1390:
1387:
1367:
1364:
1361:
1358:
1355:
1330:
1317:
1314:
1294:
1291:
1288:
1285:
1282:
1261:
1241:
1221:
1214:
1198:
1191:
1175:
1152:
1135:
1116:
1113:
1107:
1071:
1068:
1062:
1041:
1026:
1018:
1002:
999:
996:
975:
972:
969:
966:
963:
960:
957:
935:
932:
929:
926:
923:
902:
882:
862:
839:
836:
833:
823:
819:
803:
800:
797:
794:
791:
769:
766:
763:
760:
757:
736:
729:
713:
706:
687:
684:
681:
671:
667:
648:
640:
636:
617:
614:
599:
595:
579:
576:
573:
570:
567:
545:
542:
539:
536:
533:
512:
492:
484:
480:
461:
458:
443:
439:
438:
434:
416:
413:
410:
390:
387:
384:
381:
378:
370:
369:
368:
366:
350:
343:
322:
319:
316:
313:
310:
302:
301:
300:
286:
278:
262:
255:
239:
219:
207:
205:
203:
202:
197:
196:
191:
186:
184:
180:
176:
172:
168:
164:
160:
156:
150:
126:
118:
76:
70:
63:
59:
55:
51:
47:
43:
37:
33:
19:
4466:
4456:
4429:
4392:
4366:
4348:
4345:"Idempotent"
4325:
4292:
4261:the original
4256:
4244:
4231:
4215:
4207:
4185:
4175:
4166:. See also
4151:
4098:
3966:
3649:
3640:
3630:
3624:
3602:
3594:
3584:
3580:
3576:
3572:
3568:
3564:
3562:
3557:
3550:
3540:
3536:
3532:
3528:
3526:
3511:
3504:
3488:
3473:
3469:Reprinted:
3445:
3441:
3425:
3421:
3412:
3397:
3394:"idempotent"
3388:
3377:
3368:
3270:
3238:
3231:
3224:
3209:
3202:
3198:
3192:
3188:HTTP methods
3177:
3012:
3009:
3001:
2992:
2967:side effects
2950:
2944:
2643:
2628:
2622:
2399:
2390:vector space
2349:affine space
1823:
1787:Boolean ring
989:and finally
364:
341:
339:
276:
211:
199:
187:
68:
67:
61:
57:
49:
45:
4451:p. 443
4426:Lang, Serge
3527:An element
3466:See p. 104.
3193:nullipotent
2951:idempotence
2949:, the term
2937:Stable sort
2508:, and then
2400:If the set
2394:projections
2379:Kleene plus
2375:Kleene star
2345:convex hull
2116:fixed point
1805:determinant
1168:of the set
212:An element
159:mathematics
69:Idempotence
18:Idempotency
4500:Categories
4448:0848.13001
4338:0898.16032
4282:idempotent
3541:idempotent
3539:is called
3495:, p.
3434:Printed:
3426:idempotent
3361:References
3216:page fault
2963:subroutine
2927:See also:
2180:, that is
1944:such that
1750:GCD domain
915:such that
365:idempotent
277:idempotent
208:Definition
171:projectors
155:operations
4355:EMS Press
4284:" at the
4142:is not a
4079:≠
3945:∘
3930:∘
3921:∘
3912:∘
3897:∘
3888:∘
3879:∘
3867:∘
3858:∘
3849:∘
3834:∘
3825:∘
3816:∘
3785:∘
3717:∘
3705:∘
3422:nilpotent
3345:Nilpotent
2899:⋅
2893:−
2890:∘
2884:⋅
2878:−
2853:⋅
2847:−
2825:¬
2822:∘
2819:¬
2798:¬
2774:∘
2746:∘
2603:−
2553:∑
2524:−
2472:−
2351:over the
2222:
2201:
2192:
2158:∘
2099:∈
2043:∈
1956:∘
1928:→
1922::
1898:∘
1848:∘
1717:∈
1685:∧
1645:∈
1613:∨
1589:∧
1569:∨
1549:∧
1526:∨
1496:∧
1452:∨
1391:∈
1359:∩
1318:∈
1286:∪
1262:∩
1242:∪
1222:∩
1199:∪
1190:set union
1134:power set
1117:∩
1072:∪
973:⋅
961:⋅
927:⋅
840:⋅
795:⋅
761:⋅
688:⋅
663:0 + 0 = 0
571:×
537:×
462:×
414:∈
382:⋅
351:⋅
314:⋅
287:⋅
263:⋅
232:of a set
4428:(1993),
4160:Archived
3801:, since
3579:tel que
3404:Archived
3283:See also
3273:elevator
3155:sequence
3146:sequence
3104:sequence
3053:"%d
3004:database
2392:are its
2364:interior
2268:constant
2243:for all
2032:for all
1706:for all
1634:for all
1380:for all
1307:for all
639:addition
435:Examples
403:for all
4490:1896125
4430:Algebra
4419:1838439
4385:2106764
4357:, 2001
4311:1150975
3462:2369153
3255:, etc.
3178:In the
3125:inspect
3113:inspect
3038:inspect
2637:in the
2634:A000248
2360:closure
2286:ceiling
1887:) with
1511:of the
1132:of the
949:, then
641:, only
633:of the
596:In the
485:, only
477:of the
440:In the
201:potence
4488:
4478:
4446:
4436:
4417:
4407:
4383:
4373:
4336:
4309:
4299:
4155:IETF,
4029:, but
3611:
3519:
3460:
3184:safety
3164:return
3119:change
3077:change
3059:"
3047:printf
2939:, and
2684:3 and
1807:of an
1803:, the
1673:, and
1347:, and
726:or an
598:monoid
442:monoid
279:under
177:) and
4460:1870.
4322:(PDF)
4264:(PDF)
4253:(PDF)
3970:e.g.
3563:Soit
3478:(PDF)
3458:JSTOR
3210:In a
2965:with
2388:of a
2353:reals
2329:of a
2282:floor
2114:is a
1799:In a
1792:In a
1785:In a
1748:In a
1515:with
1188:with
822:group
820:In a
703:, an
670:magma
668:In a
637:with
481:with
4476:ISBN
4434:ISBN
4405:ISBN
4371:ISBN
4297:ISBN
4204:7231
4122:and
3753:and
3673:and
3609:ISBN
3517:ISBN
3137:main
3101:void
3074:void
3035:void
2981:, a
2961:, a
2639:OEIS
2420:has
2377:and
2373:the
2362:and
2358:the
2343:the
2336:the
2288:and
2280:the
2273:the
2142:the
1778:and
1581:and
1538:and
1467:and
1254:and
1211:and
1087:and
783:and
559:and
505:and
340:The
195:idem
173:and
161:and
34:and
4472:127
4444:Zbl
4397:doi
4334:Zbl
4201:RFC
4191:doi
3653:If
3575:de
3497:127
3450:doi
3243:,
3232:In
3203:In
3158:();
3149:();
3134:int
3128:();
3122:();
3116:();
3020:int
3011:on—
2977:in
2957:in
2945:In
2707:max
2682:mod
2641:).
2219:abs
2198:abs
2189:abs
2167:abs
2161:abs
2155:abs
2118:of
1780:LCM
1776:GCD
1019:of
367:if
299:if
185:).
157:in
62:Off
50:Off
4502::
4486:MR
4484:,
4474:,
4442:,
4415:MR
4413:,
4403:,
4381:MR
4379:,
4353:,
4347:,
4307:MR
4305:,
4255:.
4222:.
4206:.
4199:.
4183:.
3583:=
3535:=
3533:ss
3525:.
3456:.
3444:.
3440:.
3402:.
3396:.
3376:.
3140:()
3107:()
3080:()
3068:);
3056:\n
3041:()
2935:,
2931:,
2284:,
198:+
149:-/
137:aɪ
127::
125:US
121:,
111:ən
105:oʊ
77::
75:UK
58:On
46:On
4399::
4280:"
4226:.
4193::
4170:.
4130:g
4110:f
4082:1
4076:2
4073:=
4070:)
4067:5
4064:(
4061:f
4058:=
4055:)
4052:)
4049:1
4046:(
4043:g
4040:(
4037:f
4017:1
4014:=
4011:)
4008:7
4005:(
4002:f
3999:=
3996:)
3993:)
3990:7
3987:(
3984:g
3981:(
3978:f
3948:g
3942:f
3939:=
3936:)
3933:g
3927:g
3924:(
3918:)
3915:f
3909:f
3906:(
3903:=
3900:g
3894:)
3891:g
3885:f
3882:(
3876:f
3873:=
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