1470:
458:
is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean 'category in which all morphisms are invertible'. The term
880:
The magma operation may be applied repeatedly, and in the general, non-associative case, the order matters, which is notated with parentheses. Also, the operation • is often omitted and notated by juxtaposition:
847:
1477:
Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:
636:
919:
A shorthand is often used to reduce the number of parentheses, in which the innermost operations and pairs of parentheses are omitted, being replaced just with juxtaposition:
454:
According to
Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The word
435:
in the sense used in category theory, but not in the sense used by
Hausmann and Ore. Nevertheless, influential books in semigroup theory, including
358:
780:
1227:
1203:
1179:
3187:
3673:
3389:
3359:
3330:
3303:
3228:
3520:
427:(1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in
530:. The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation
351:
3564:
3476:
3414:
1459:
586:
3691:
1267:
is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example:
474:
3464:
3701:
3627:
3609:
3591:
3493:
3468:
3176:
344:
3622:
3604:
3586:
436:
213:
3696:
400:
3056:
1008:
consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the
1005:
2600:
3277:
3077:
2582:
941:. For example, the above is abbreviated to the following expression, still containing parentheses:
3127:
983:
304:
3617:
1062:
are the only two ways of pairing three elements of a magma with two operations. Less trivially,
451:
is "perhaps most often used in modern mathematics" in the sense given to it in category theory.
3040:
1665:
1487:
24:
3599:
3456:
3293:
1374:. The operation is that of joining trees at the root. It therefore has a foundational role in
3554:
3379:
3320:
2661:
1609:
1594:
1539:
743:
500:
444:
3347:
3218:
3661:
3581:
3539:
2656:
2651:
1614:
1599:
447:(1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term
431:, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a
291:
283:
255:
250:
241:
198:
140:
8:
3108:
2047:
1529:
388:
309:
299:
150:
50:
42:
33:
3260:
3146:
2025:
1565:
1382:
115:
106:
64:
3669:
3635:
3560:
3472:
3431:
3410:
3385:
3355:
3326:
3299:
3271:
3224:
3182:
3171:
3154:
3116:
3060:
2627:
2378:
2257:
1557:
861:
642:
489:
464:
392:
19:
This article is about the algebraic structure. For groupoids in category theory, see
3295:
Mathematics across the Iron
Curtain: A History of the Algebraic Theory of Semigroups
3252:
3192:
3073:
3066:
2386:
1621:
1604:
1499:
1363:
979:
853:
658:
543:
496:
468:
396:
372:
135:
3501:
Acta
Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
160:
3535:
3381:
Associahedra, Tamari
Lattices and Related Structures: Tamari Memorial Festschrift
3134:
1182:) magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-
964:
755:
420:
416:
227:
221:
208:
188:
179:
145:
2620:
If any triple of (not necessarily distinct) elements generates a medial submagma
3166:
2471:
1890:
1867:
1493:
1447:
1207:
1019:
747:
269:
3638:
3434:
424:
3685:
3405:
Evseev, A. E. (1988), "A survey of partial groupoids", in Silver, Ben (ed.),
2335:
2069:
1573:
1009:
440:
155:
120:
77:
852:
Note that these commutative magmas are not associative; nor do they have an
3120:
3104:
3100:
2631:
2495:
2128:
1845:
1822:
1643:
428:
329:
260:
94:
3378:
Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012),
2646:
2411:
1547:
1513:
1367:
1257:
319:
314:
203:
193:
167:
3243:
Hausmann, B. A.; Ore, Øystein (October 1937), "Theory of quasi-groups",
3264:
2274:
1981:
1958:
1800:
1777:
1482:
1451:
1256:(i.e., there are no relations or axioms imposed on the generators; see
1183:
69:
3643:
3439:
3097:
2435:
2291:
1935:
1912:
1508:
865:
857:
759:
324:
130:
87:
55:
3256:
3197:
3044:
1710:
1687:
1455:
998:
670:
432:
125:
20:
1469:
3112:
842:{\displaystyle \log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}}
2638:
2003:
1534:: A magma with inverse, associativity, and an identity element.
1520:
1375:
59:
3556:
Mal'cev, protomodular, homological and semi-abelian categories
3348:"Algebraic Structures: §1.1 Laws of Composition: Definition 1"
997:, in which the order of execution is simply left-to-right (no
423:. The term was then appropriated by B. A. Hausmann and
3377:
3001:
2995:
2918:
2912:
2893:
2872:
2867:
2812:
2807:
2788:
2782:
2763:
2758:
2739:
2734:
2715:
2710:
2691:
2686:
1538:
Note that each of divisibility and invertibility imply the
1222:
1206:) and the numbers of simultaneously non-isomorphic and non-
1198:
1174:
3373:
3371:
3322:
Cogroups and Co-rings in
Categories of Associative Rings
1355:
can be described as the set of non-associative words on
2383:
If the submagma generated by any element is associative
3368:
2958:
a(n)=1 for n=0 and all odd n, a(n)=0 for all even n≥2
2578:
If it is both right-cancellative and left-cancellative
783:
589:
538:
must satisfy the following requirement (known as the
3312:
3188:
Algebraic structures whose axioms are all identities
963:
A way to avoid completely the use of parentheses is
3528:
3298:, American Mathematical Society, pp. 142–143,
3220:
Universal
Algebra: Fundamentals and Selected Topics
2605:If it is a semigroup and it satisfies the identity
2587:If it is a semigroup and it satisfies the identity
1018:applications of the magma operator is given by the
1252:is the "most general possible" magma generated by
841:
630:
3633:
3429:
3318:
986:), in which the same expression would be written
3683:
3319:Bergman, George M.; Hausknecht, Adam O. (1996),
1012:. The total number of different ways of writing
967:, in which the same expression would be written
403:by definition. No other properties are imposed.
1581:
870:A Serious Fall in the Value of Gold Ascertained
631:{\displaystyle a,b\in M\implies a\cdot b\in M.}
3325:, American Mathematical Society, p. 61,
875:
352:
3552:
2639:Number of magmas satisfying given properties
1362:It can also be viewed, in terms familiar in
754:equal to the real number line, and • as the
472:
3553:Borceux, Francis; Bourn, Dominique (2004).
3339:
3285:
3242:
978:. Another way, familiar to programmers, is
856:. This morphism of magmas has been used in
16:Algebraic structure with a binary operation
3615:
3597:
3579:
3518:
3491:
1473:Algebraic structures from magmas to groups
609:
605:
406:
359:
345:
3179:, named after the object of this article.
2254:If it is both left and right distributive
3345:
3291:
1468:
1159:elements, so there are 1, 1, 16, 19683,
3407:Nineteen Papers on Algebraic Semigroups
3216:
3048:
2470:Semigroup with zero multiplication, or
2202:If it is both left and right semimedial
1524:: A semigroup with an identity element.
3684:
3519:Ježek, Jaroslav; Kepka, Tomáš (1981),
3404:
1407:, then there is a unique extension of
391:. Specifically, a magma consists of a
3660:
3634:
3461:Graduate Algebra: Noncommutative View
3454:
3430:
3398:
3030:
703:that preserves the binary operation:
664:
2669:OEIS sequence (isomorphism classes)
1370:with leaves labelled by elements of
868:in 39 commodities in England in his
3043:whose objects are magmas and whose
1040:, which is just the statement that
13:
3654:
14:
3713:
3409:, American Mathematical Society,
3096:An important property is that an
1512:: A magma where the operation is
1464:
3035:The category of magmas, denoted
3024:a(0)=a(1)=1, a(n)=0 for all n≥2
2978:a(0)=a(1)=1, a(n)=0 for all n≥2
477:, Algèbre, chapitres 1 à 3, 1970
3546:
3512:
3485:
3465:Graduate Studies in Mathematics
3245:American Journal of Mathematics
3021:a(0)=a(1)=1, a(n)=0 for all n≥2
2975:a(0)=a(1)=1, a(n)=0 for all n≥2
2340:If it satisfies the identities
2311:If it satisfies the identities
3448:
3423:
3292:Hollings, Christopher (2014),
3236:
3210:
1569:: A monoid with commutativity.
606:
580:And in mathematical notation:
562:, the result of the operation
1:
3492:Kepka, T.; Němec, P. (1996),
3469:American Mathematical Society
3203:
3177:Magma computer algebra system
2491:If it has an identity element
2476:If it satisfies the identity
2458:If it satisfies the identity
2443:If it satisfies the identity
2416:If it satisfies the identity
2296:If it satisfies the identity
2279:If it satisfies the identity
2262:If it satisfies the identity
2231:If it satisfies the identity
2208:If it satisfies the identity
2179:If it satisfies the identity
2156:If it satisfies the identity
2133:If it satisfies the identity
1577:: A group with commutativity.
1561:: A magma with commutativity.
1460:Wedderburn–Etherington number
1233:
483:
3559:. Springer. pp. 7, 19.
1980:Commutative-and-associative
1582:Classification by properties
765:is a morphism of the magma (
7:
3623:Encyclopedia of Mathematics
3605:Encyclopedia of Mathematics
3587:Encyclopedia of Mathematics
3494:"Simple balanced groupoids"
3455:Rowen, Louis Halle (2008),
3160:
1359:with parentheses retained.
1260:). The binary operation on
1186:magmas are 1, 1, 10, 3330,
10:
3718:
3668:(3rd ed.), Springer,
3666:A survey of binary systems
3217:Bergman, Clifford (2011),
2601:semigroup with right zeros
1445:
1210:magmas are 1, 1, 7, 1734,
876:Notation and combinatorics
415:was introduced in 1927 by
18:
3521:"Free entropic groupoids"
2921:with a(0)=1 instead of 0
2583:semigroup with left zeros
3616:Hazewinkel, M. (2001) ,
3598:Hazewinkel, M. (2001) ,
3580:Hazewinkel, M. (2001) ,
3384:, Springer, p. 11,
3072:as trivial magmas, with
2666:OEIS sequence (labeled)
1411:to a morphism of magmas
1004:The set of all possible
673:of magmas is a function
475:Éléments de mathématique
3692:Non-associative algebra
3354:, Springer, p. 1,
3352:Algebra I: Chapters 1–3
1498:: A quasigroup with an
1366:, as the magma of full
984:reverse Polish notation
864:calculated the rate of
407:History and terminology
395:equipped with a single
3346:Bourbaki, N. (1998) ,
3103:can be extended to an
1587:Group-like structures
1474:
843:
632:
473:
467:." It also appears in
25:Magma (disambiguation)
23:. For other uses, see
3662:Bruck, Richard Hubert
2662:Cancellation property
1540:cancellation property
1472:
1381:A free magma has the
1220:, ... (sequence
1196:, ... (sequence
1172:, ... (sequence
1030:. Thus, for example,
844:
744:positive real numbers
633:
499:• that sends any two
3702:Algebraic structures
3276:: CS1 maint: year (
2657:Associative property
2652:Commutative property
1490:is always possible.
781:
587:
516:to another element,
256:Group with operators
199:Complemented lattice
34:Algebraic structures
3457:"Definition 21B.1."
3049:magma homomorphisms
1588:
1399:is a function from
389:algebraic structure
387:is a basic kind of
310:Composition algebra
70:Quasigroup and loop
3636:Weisstein, Eric W.
3432:Weisstein, Eric W.
3059:, and there is an
3031:Category of magmas
2536:Right-cancellative
2228:Right distributive
2026:Commutative monoid
1586:
1566:Commutative monoid
1475:
1383:universal property
839:
738:For example, with
665:Morphism of magmas
657:or, more often, a
641:If • is instead a
628:
3697:Binary operations
3675:978-0-387-03497-3
3391:978-3-0348-0405-9
3361:978-3-540-64243-5
3332:978-0-8218-0495-7
3305:978-1-4704-1493-1
3230:978-1-4398-5130-2
3183:Commutative magma
3172:Universal algebra
3119:sequence of the)
3061:inclusion functor
3028:
3027:
2628:homomorphic image
2379:Power-associative
2205:Left distributive
2090:
2089:
1558:Commutative magma
862:W. Stanley Jevons
837:
808:
802:
798:
643:partial operation
369:
368:
3709:
3678:
3649:
3648:
3630:
3612:
3594:
3571:
3570:
3550:
3544:
3542:
3525:
3516:
3510:
3508:
3498:
3489:
3483:
3481:
3452:
3446:
3445:
3444:
3427:
3421:
3419:
3402:
3396:
3394:
3375:
3366:
3364:
3343:
3337:
3335:
3316:
3310:
3308:
3289:
3283:
3281:
3275:
3267:
3240:
3234:
3233:
3214:
3193:Groupoid algebra
3132:
3092:
3071:
2643:
2642:
2614:
2596:
2572:
2562:
2552:
2533:
2523:
2513:
2485:
2467:
2452:
2433:
2408:
2375:
2357:
2332:
2305:
2288:
2271:
2251:Autodistributive
2248:
2225:
2196:
2176:Right semimedial
2173:
2150:
2123:
2117:
2099:
1589:
1585:
1500:identity element
1486:: A magma where
1430:
1427:
1419:
1416:
1398:
1364:computer science
1345:
1316:
1288:
1225:
1219:
1218:
1215:
1201:
1195:
1194:
1191:
1177:
1171:
1170:
1167:
1164:
1158:
1152:
1143:
1128:
1113:
1101:
1086:
1071:
1061:
1050:
1039:
1029:
1017:
996:
980:postfix notation
977:
958:
940:
914:
860:since 1863 when
854:identity element
848:
846:
845:
840:
838:
833:
810:
806:
800:
799:
791:
702:
694:
687:that maps magma
686:
659:partial groupoid
652:
637:
635:
634:
629:
571:
544:closure property
537:
529:
515:
495:matched with an
479:
397:binary operation
373:abstract algebra
361:
354:
347:
136:Commutative ring
65:Rack and quandle
30:
29:
3717:
3716:
3712:
3711:
3710:
3708:
3707:
3706:
3682:
3681:
3676:
3657:
3655:Further reading
3652:
3575:
3574:
3567:
3551:
3547:
3523:
3517:
3513:
3496:
3490:
3486:
3479:
3471:, p. 321,
3453:
3449:
3428:
3424:
3417:
3403:
3399:
3392:
3376:
3369:
3362:
3344:
3340:
3333:
3317:
3313:
3306:
3290:
3286:
3269:
3268:
3257:10.2307/2371362
3251:(4): 983–1004,
3241:
3237:
3231:
3215:
3211:
3206:
3163:
3153:is pointed and
3135:terminal object
3130:
3080:
3064:
3057:direct products
3051:. The category
3033:
2641:
2606:
2588:
2564:
2554:
2540:
2525:
2515:
2501:
2477:
2459:
2444:
2417:
2392:
2359:
2341:
2312:
2297:
2280:
2263:
2232:
2209:
2180:
2157:
2153:Left semimedial
2134:
2119:
2101:
2093:
1584:
1467:
1462:
1436:
1428:
1423:
1417:
1412:
1386:
1353:
1319:
1291:
1271:
1265:
1246:
1236:
1221:
1216:
1213:
1211:
1197:
1192:
1189:
1187:
1173:
1168:
1165:
1162:
1160:
1154:
1148:
1130:
1115:
1103:
1088:
1073:
1069:
1063:
1052:
1041:
1037:
1031:
1027:
1022:
1013:
987:
968:
965:prefix notation
945:
920:
885:
878:
811:
809:
790:
782:
779:
778:
756:arithmetic mean
696:
688:
674:
667:
646:
588:
585:
584:
563:
531:
517:
503:
486:
421:Brandt groupoid
419:describing his
417:Heinrich Brandt
409:
365:
336:
335:
334:
305:Non-associative
287:
276:
275:
265:
245:
234:
233:
222:Map of lattices
218:
214:Boolean algebra
209:Heyting algebra
183:
172:
171:
165:
146:Integral domain
110:
99:
98:
92:
46:
28:
17:
12:
11:
5:
3715:
3705:
3704:
3699:
3694:
3680:
3679:
3674:
3656:
3653:
3651:
3650:
3631:
3613:
3595:
3576:
3573:
3572:
3565:
3545:
3534:(2): 223–233,
3511:
3484:
3477:
3447:
3422:
3415:
3397:
3390:
3367:
3360:
3338:
3331:
3311:
3304:
3284:
3235:
3229:
3208:
3207:
3205:
3202:
3201:
3200:
3195:
3190:
3185:
3180:
3174:
3169:
3167:Magma category
3162:
3159:
3141:, and because
3032:
3029:
3026:
3025:
3022:
3019:
3016:
3013:
3010:
3006:
3005:
2999:
2993:
2990:
2987:
2984:
2980:
2979:
2976:
2973:
2970:
2967:
2964:
2960:
2959:
2956:
2954:
2951:
2948:
2945:
2941:
2940:
2938:
2936:
2933:
2930:
2927:
2923:
2922:
2916:
2910:
2907:
2904:
2901:
2897:
2896:
2891:
2889:
2886:
2883:
2880:
2876:
2875:
2870:
2865:
2862:
2859:
2856:
2852:
2851:
2849:
2847:
2844:
2841:
2838:
2834:
2833:
2831:
2829:
2826:
2823:
2820:
2816:
2815:
2810:
2805:
2802:
2799:
2796:
2792:
2791:
2786:
2780:
2777:
2774:
2771:
2767:
2766:
2761:
2756:
2753:
2750:
2747:
2743:
2742:
2737:
2732:
2729:
2726:
2723:
2719:
2718:
2713:
2708:
2705:
2702:
2699:
2695:
2694:
2689:
2684:
2681:
2678:
2675:
2671:
2670:
2667:
2664:
2659:
2654:
2649:
2640:
2637:
2636:
2635:
2624:
2621:
2618:
2615:
2603:
2597:
2585:
2579:
2576:
2573:
2537:
2534:
2498:
2492:
2489:
2486:
2474:
2472:null semigroup
2468:
2456:
2453:
2441:
2438:
2414:
2409:
2389:
2384:
2381:
2376:
2338:
2333:
2309:
2306:
2294:
2289:
2277:
2272:
2260:
2255:
2252:
2249:
2229:
2226:
2206:
2203:
2200:
2197:
2177:
2174:
2154:
2151:
2131:
2088:
2087:
2084:
2081:
2078:
2075:
2072:
2066:
2065:
2062:
2059:
2056:
2053:
2050:
2044:
2043:
2040:
2037:
2034:
2031:
2028:
2022:
2021:
2018:
2015:
2012:
2009:
2006:
2000:
1999:
1996:
1993:
1990:
1987:
1984:
1977:
1976:
1973:
1970:
1967:
1964:
1961:
1954:
1953:
1950:
1947:
1944:
1941:
1938:
1931:
1930:
1927:
1924:
1921:
1918:
1915:
1909:
1908:
1905:
1902:
1899:
1896:
1893:
1886:
1885:
1882:
1879:
1876:
1873:
1870:
1864:
1863:
1860:
1857:
1854:
1851:
1848:
1841:
1840:
1837:
1834:
1831:
1828:
1825:
1819:
1818:
1815:
1812:
1809:
1806:
1803:
1796:
1795:
1792:
1789:
1786:
1783:
1780:
1774:
1773:
1770:
1767:
1764:
1761:
1758:
1751:
1750:
1747:
1744:
1741:
1738:
1735:
1729:
1728:
1725:
1722:
1719:
1716:
1713:
1706:
1705:
1702:
1699:
1696:
1693:
1690:
1684:
1683:
1680:
1677:
1674:
1671:
1668:
1666:Small category
1662:
1661:
1658:
1655:
1652:
1649:
1646:
1640:
1639:
1636:
1633:
1630:
1627:
1624:
1618:
1617:
1612:
1607:
1602:
1597:
1592:
1583:
1580:
1579:
1578:
1570:
1562:
1553:
1552:
1550:
1536:
1535:
1527:
1526:
1525:
1505:
1504:
1503:
1466:
1465:Types of magma
1463:
1448:Free semigroup
1444:
1443:
1434:
1351:
1347:
1346:
1317:
1289:
1263:
1244:
1235:
1232:
1208:antiisomorphic
1067:
1035:
1025:
1020:Catalan number
961:
960:
917:
916:
877:
874:
850:
849:
836:
832:
829:
826:
823:
820:
817:
814:
805:
797:
794:
789:
786:
748:geometric mean
736:
735:
666:
663:
639:
638:
627:
624:
621:
618:
615:
612:
608:
604:
601:
598:
595:
592:
578:
577:
485:
482:
408:
405:
383:, or, rarely,
367:
366:
364:
363:
356:
349:
341:
338:
337:
333:
332:
327:
322:
317:
312:
307:
302:
296:
295:
294:
288:
282:
281:
278:
277:
274:
273:
270:Linear algebra
264:
263:
258:
253:
247:
246:
240:
239:
236:
235:
232:
231:
228:Lattice theory
224:
217:
216:
211:
206:
201:
196:
191:
185:
184:
178:
177:
174:
173:
164:
163:
158:
153:
148:
143:
138:
133:
128:
123:
118:
112:
111:
105:
104:
101:
100:
91:
90:
85:
80:
74:
73:
72:
67:
62:
53:
47:
41:
40:
37:
36:
15:
9:
6:
4:
3:
2:
3714:
3703:
3700:
3698:
3695:
3693:
3690:
3689:
3687:
3677:
3671:
3667:
3663:
3659:
3658:
3646:
3645:
3640:
3637:
3632:
3629:
3625:
3624:
3619:
3614:
3611:
3607:
3606:
3601:
3596:
3593:
3589:
3588:
3583:
3578:
3577:
3568:
3566:1-4020-1961-0
3562:
3558:
3557:
3549:
3541:
3537:
3533:
3529:
3522:
3515:
3506:
3502:
3495:
3488:
3480:
3478:0-8218-8408-5
3474:
3470:
3466:
3462:
3458:
3451:
3442:
3441:
3436:
3433:
3426:
3418:
3416:0-8218-3115-1
3412:
3408:
3401:
3393:
3387:
3383:
3382:
3374:
3372:
3363:
3357:
3353:
3349:
3342:
3334:
3328:
3324:
3323:
3315:
3307:
3301:
3297:
3296:
3288:
3279:
3273:
3266:
3262:
3258:
3254:
3250:
3246:
3239:
3232:
3226:
3223:, CRC Press,
3222:
3221:
3213:
3209:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3178:
3175:
3173:
3170:
3168:
3165:
3164:
3158:
3156:
3152:
3148:
3144:
3140:
3136:
3129:
3124:
3122:
3118:
3114:
3110:
3106:
3102:
3099:
3094:
3091:
3087:
3083:
3079:
3075:
3070:
3068:
3062:
3058:
3054:
3050:
3046:
3042:
3038:
3023:
3020:
3017:
3014:
3011:
3008:
3007:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2981:
2977:
2974:
2971:
2968:
2965:
2962:
2961:
2957:
2955:
2952:
2949:
2946:
2943:
2942:
2939:
2937:
2934:
2931:
2928:
2925:
2924:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2898:
2895:
2892:
2890:
2887:
2884:
2881:
2878:
2877:
2874:
2871:
2869:
2866:
2863:
2860:
2857:
2854:
2853:
2850:
2848:
2845:
2842:
2839:
2836:
2835:
2832:
2830:
2827:
2824:
2821:
2818:
2817:
2814:
2811:
2809:
2806:
2803:
2800:
2797:
2794:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2768:
2765:
2762:
2760:
2757:
2754:
2751:
2748:
2745:
2744:
2741:
2738:
2736:
2733:
2730:
2727:
2724:
2721:
2720:
2717:
2714:
2712:
2709:
2706:
2703:
2700:
2697:
2696:
2693:
2690:
2688:
2685:
2682:
2679:
2676:
2673:
2672:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2644:
2633:
2629:
2625:
2622:
2619:
2616:
2613:
2609:
2604:
2602:
2598:
2595:
2591:
2586:
2584:
2580:
2577:
2574:
2571:
2567:
2561:
2557:
2551:
2547:
2543:
2538:
2535:
2532:
2528:
2522:
2518:
2512:
2508:
2504:
2499:
2497:
2493:
2490:
2487:
2484:
2480:
2475:
2473:
2469:
2466:
2462:
2457:
2454:
2451:
2447:
2442:
2439:
2437:
2432:
2428:
2424:
2420:
2415:
2413:
2410:
2407:
2403:
2399:
2395:
2390:
2388:
2385:
2382:
2380:
2377:
2374:
2370:
2366:
2362:
2356:
2352:
2348:
2344:
2339:
2337:
2334:
2331:
2327:
2323:
2319:
2315:
2310:
2307:
2304:
2300:
2295:
2293:
2290:
2287:
2283:
2278:
2276:
2273:
2270:
2266:
2261:
2259:
2256:
2253:
2250:
2247:
2243:
2239:
2235:
2230:
2227:
2224:
2220:
2216:
2212:
2207:
2204:
2201:
2198:
2195:
2191:
2187:
2183:
2178:
2175:
2172:
2168:
2164:
2160:
2155:
2152:
2149:
2145:
2141:
2137:
2132:
2130:
2127:
2126:
2125:
2122:
2116:
2112:
2108:
2104:
2097:
2085:
2082:
2079:
2076:
2073:
2071:
2070:Abelian group
2068:
2067:
2063:
2060:
2057:
2054:
2051:
2049:
2046:
2045:
2041:
2038:
2035:
2032:
2029:
2027:
2024:
2023:
2019:
2016:
2013:
2010:
2007:
2005:
2002:
2001:
1997:
1994:
1991:
1988:
1985:
1983:
1979:
1978:
1974:
1971:
1968:
1965:
1962:
1960:
1956:
1955:
1951:
1948:
1945:
1942:
1939:
1937:
1933:
1932:
1928:
1925:
1922:
1919:
1916:
1914:
1911:
1910:
1906:
1903:
1900:
1897:
1894:
1892:
1888:
1887:
1883:
1880:
1877:
1874:
1871:
1869:
1866:
1865:
1861:
1858:
1855:
1852:
1849:
1847:
1843:
1842:
1838:
1835:
1832:
1829:
1826:
1824:
1821:
1820:
1816:
1813:
1810:
1807:
1804:
1802:
1798:
1797:
1793:
1790:
1787:
1784:
1781:
1779:
1776:
1775:
1771:
1768:
1765:
1762:
1759:
1757:
1753:
1752:
1748:
1745:
1742:
1739:
1736:
1734:
1731:
1730:
1726:
1723:
1720:
1717:
1714:
1712:
1708:
1707:
1703:
1700:
1697:
1694:
1691:
1689:
1686:
1685:
1681:
1678:
1675:
1672:
1669:
1667:
1664:
1663:
1659:
1656:
1653:
1650:
1647:
1645:
1642:
1641:
1637:
1634:
1631:
1628:
1625:
1623:
1622:Partial magma
1620:
1619:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1590:
1576:
1575:
1574:Abelian group
1571:
1568:
1567:
1563:
1560:
1559:
1555:
1554:
1551:
1549:
1548:commutativity
1545:
1544:
1543:
1541:
1533:
1532:
1528:
1523:
1522:
1518:
1517:
1515:
1511:
1510:
1506:
1501:
1497:
1496:
1492:
1491:
1489:
1485:
1484:
1480:
1479:
1478:
1471:
1461:
1457:
1453:
1449:
1441:
1437:
1426:
1422:
1421:
1420:
1415:
1410:
1406:
1403:to any magma
1402:
1397:
1393:
1389:
1385:such that if
1384:
1379:
1377:
1373:
1369:
1365:
1360:
1358:
1354:
1343:
1339:
1335:
1331:
1327:
1323:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1269:
1268:
1266:
1259:
1255:
1251:
1247:
1241:
1231:
1229:
1224:
1209:
1205:
1200:
1185:
1181:
1176:
1157:
1151:
1145:
1141:
1137:
1133:
1126:
1122:
1118:
1111:
1107:
1100:
1096:
1092:
1085:
1081:
1077:
1066:
1059:
1055:
1049:
1045:
1034:
1028:
1021:
1016:
1011:
1010:Dyck language
1007:
1002:
1000:
994:
990:
985:
981:
976:
972:
966:
957:
953:
949:
944:
943:
942:
939:
935:
931:
927:
923:
913:
909:
905:
901:
897:
893:
889:
884:
883:
882:
873:
871:
867:
863:
859:
855:
834:
830:
827:
824:
821:
818:
815:
812:
803:
795:
792:
787:
784:
776:
775:
774:
772:
768:
764:
761:
757:
753:
749:
746:and * as the
745:
742:equal to the
741:
733:
729:
725:
721:
717:
713:
709:
706:
705:
704:
700:
692:
685:
681:
677:
672:
662:
660:
656:
655:partial magma
650:
644:
625:
622:
619:
616:
613:
610:
602:
599:
596:
593:
590:
583:
582:
581:
575:
570:
566:
561:
557:
553:
549:
548:
547:
545:
541:
535:
528:
524:
520:
514:
510:
506:
502:
498:
494:
491:
488:A magma is a
481:
478:
476:
470:
466:
462:
457:
452:
450:
446:
442:
438:
434:
430:
426:
422:
418:
414:
404:
402:
399:that must be
398:
394:
390:
386:
382:
378:
374:
362:
357:
355:
350:
348:
343:
342:
340:
339:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
293:
290:
289:
285:
280:
279:
272:
271:
267:
266:
262:
259:
257:
254:
252:
249:
248:
243:
238:
237:
230:
229:
225:
223:
220:
219:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
181:
176:
175:
170:
169:
162:
159:
157:
156:Division ring
154:
152:
149:
147:
144:
142:
139:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
113:
108:
103:
102:
97:
96:
89:
86:
84:
81:
79:
78:Abelian group
76:
75:
71:
68:
66:
63:
61:
57:
54:
52:
49:
48:
44:
39:
38:
35:
32:
31:
26:
22:
3665:
3642:
3621:
3618:"Free magma"
3603:
3585:
3555:
3548:
3531:
3527:
3514:
3504:
3500:
3487:
3460:
3450:
3438:
3425:
3406:
3400:
3380:
3351:
3341:
3321:
3314:
3294:
3287:
3248:
3244:
3238:
3219:
3212:
3150:
3142:
3138:
3126:Because the
3125:
3121:endomorphism
3105:automorphism
3101:endomorphism
3095:
3089:
3085:
3081:
3065:
3052:
3036:
3034:
2632:cancellation
2630:of a medial
2611:
2607:
2593:
2589:
2575:Cancellative
2569:
2565:
2559:
2555:
2549:
2545:
2541:
2539:If, for all
2530:
2526:
2520:
2516:
2510:
2506:
2502:
2500:If, for all
2496:cancellative
2482:
2478:
2464:
2460:
2455:A right unar
2449:
2445:
2430:
2426:
2422:
2418:
2405:
2401:
2397:
2393:
2372:
2368:
2364:
2360:
2354:
2350:
2346:
2342:
2329:
2325:
2321:
2317:
2313:
2302:
2298:
2285:
2281:
2268:
2264:
2245:
2241:
2237:
2233:
2222:
2218:
2214:
2210:
2193:
2189:
2185:
2181:
2170:
2166:
2162:
2158:
2147:
2143:
2139:
2135:
2124:, is called
2120:
2114:
2110:
2106:
2102:
2095:
2091:
1957:Associative
1934:Commutative
1889:Commutative
1846:unital magma
1844:Commutative
1823:Unital magma
1799:Commutative
1755:
1754:Commutative
1732:
1709:Commutative
1644:Semigroupoid
1610:Cancellation
1572:
1564:
1556:
1546:Magmas with
1537:
1530:
1519:
1507:
1494:
1481:
1476:
1439:
1432:
1424:
1413:
1408:
1404:
1400:
1395:
1391:
1387:
1380:
1371:
1368:binary trees
1361:
1356:
1349:
1348:
1341:
1337:
1333:
1329:
1325:
1321:
1312:
1308:
1304:
1300:
1296:
1292:
1284:
1280:
1276:
1272:
1261:
1253:
1249:
1242:
1239:
1237:
1155:
1153:magmas with
1149:
1146:
1139:
1135:
1131:
1124:
1120:
1116:
1109:
1105:
1098:
1094:
1090:
1083:
1079:
1075:
1064:
1057:
1053:
1047:
1043:
1032:
1023:
1014:
1003:
992:
988:
974:
970:
962:
955:
951:
947:
937:
933:
929:
925:
921:
918:
911:
907:
903:
899:
895:
891:
887:
879:
869:
851:
770:
766:
762:
751:
739:
737:
731:
727:
723:
719:
715:
711:
707:
698:
690:
683:
679:
675:
668:
654:
653:is called a
648:
640:
579:
573:
568:
564:
559:
555:
551:
539:
533:
526:
522:
518:
512:
508:
504:
492:
487:
463:was used by
460:
455:
453:
448:
429:Zentralblatt
412:
410:
384:
380:
376:
370:
330:Hopf algebra
268:
261:Vector space
226:
166:
95:Group theory
93:
82:
58: /
3111:, just the
3107:of a magma
3069:→ Med ↪ Mag
3004:add a(0)=1
2647:Idempotence
2626:If it is a
2553:, relation
2514:, relation
2440:A left unar
2434:, called a
2412:Associative
2336:Alternative
2258:Commutative
1615:Commutative
1600:Associative
1514:associative
1258:free object
572:is also in
443:(1961) and
425:Øystein Ore
315:Lie algebra
300:Associative
204:Total order
194:Semilattice
168:Ring theory
3686:Categories
3639:"Groupoid"
3600:"Groupoid"
3507:(1): 53–60
3435:"Groupoid"
3204:References
3078:projection
3074:operations
2998:add a(0)=1
2915:add a(0)=1
2785:add a(0)=1
2308:Zeropotent
2275:Idempotent
2199:Semimedial
1982:quasigroup
1959:quasigroup
1801:quasigroup
1778:Quasigroup
1483:Quasigroup
1452:Free group
1446:See also:
1240:free magma
1234:Free magma
1184:isomorphic
1147:There are
872:, page 7.
484:Definition
3644:MathWorld
3628:EMS Press
3610:EMS Press
3592:EMS Press
3440:MathWorld
3147:algebraic
3128:singleton
3109:extension
3098:injective
3076:given by
3045:morphisms
3039:, is the
2617:Trimedial
2436:semigroup
2292:Unipotent
2086:Required
2064:Unneeded
2042:Required
2020:Unneeded
1998:Required
1975:Unneeded
1952:Required
1936:semigroup
1929:Unneeded
1913:Semigroup
1907:Required
1884:Unneeded
1862:Required
1839:Unneeded
1817:Required
1794:Unneeded
1772:Required
1749:Unneeded
1727:Required
1704:Unneeded
1682:Unneeded
1660:Unneeded
1638:Unneeded
1509:Semigroup
1248:on a set
866:inflation
858:economics
828:
816:
788:
769:, *) to (
760:logarithm
695:to magma
620:∈
614:⋅
607:⟹
600:∈
497:operation
411:The term
325:Bialgebra
131:Near-ring
88:Lie group
56:Semigroup
3664:(1971),
3272:citation
3198:Hall set
3161:See also
3155:complete
3131:({*}, *)
3117:constant
3115:of the (
3041:category
3018:Required
3015:Required
3012:Required
3009:Required
2992:Required
2989:Required
2986:Required
2983:Unneeded
2972:Required
2969:Required
2966:Unneeded
2963:Required
2953:Required
2950:Unneeded
2947:Required
2944:Required
2935:Unneeded
2932:Required
2929:Required
2926:Required
2909:Required
2906:Required
2903:Unneeded
2900:Unneeded
2888:Required
2885:Unneeded
2882:Required
2879:Unneeded
2864:Unneeded
2861:Required
2858:Required
2855:Unneeded
2846:Required
2843:Unneeded
2840:Unneeded
2837:Required
2828:Unneeded
2825:Required
2822:Unneeded
2819:Required
2804:Unneeded
2801:Unneeded
2798:Required
2795:Required
2779:Required
2776:Unneeded
2773:Unneeded
2770:Unneeded
2755:Unneeded
2752:Required
2749:Unneeded
2746:Unneeded
2731:Unneeded
2728:Unneeded
2725:Required
2722:Unneeded
2707:Unneeded
2704:Unneeded
2701:Unneeded
2698:Required
2683:Unneeded
2680:Unneeded
2677:Unneeded
2674:Unneeded
2623:Entropic
2563:implies
2524:implies
2387:Flexible
2092:A magma
2083:Required
2080:Required
2077:Required
2074:Required
2061:Required
2058:Required
2055:Required
2052:Required
2039:Unneeded
2036:Required
2033:Required
2030:Required
2017:Unneeded
2014:Required
2011:Required
2008:Required
1995:Required
1992:Unneeded
1989:Required
1986:Required
1972:Required
1969:Unneeded
1966:Required
1963:Required
1949:Unneeded
1946:Unneeded
1943:Required
1940:Required
1926:Unneeded
1923:Unneeded
1920:Required
1917:Required
1904:Required
1901:Required
1898:Unneeded
1895:Required
1881:Required
1878:Required
1875:Unneeded
1872:Required
1859:Unneeded
1856:Required
1853:Unneeded
1850:Required
1836:Unneeded
1833:Required
1830:Unneeded
1827:Required
1814:Required
1811:Unneeded
1808:Unneeded
1805:Required
1791:Required
1788:Unneeded
1785:Unneeded
1782:Required
1769:Unneeded
1766:Unneeded
1763:Unneeded
1760:Required
1746:Unneeded
1743:Unneeded
1740:Unneeded
1737:Required
1724:Required
1721:Required
1718:Required
1715:Unneeded
1711:Groupoid
1701:Required
1698:Required
1695:Required
1692:Unneeded
1688:Groupoid
1679:Unneeded
1676:Required
1673:Required
1670:Unneeded
1657:Unneeded
1654:Unneeded
1651:Required
1648:Unneeded
1635:Unneeded
1632:Unneeded
1629:Unneeded
1626:Unneeded
1605:Identity
1488:division
1456:Hall set
1431: :
1390: :
999:currying
678: :
671:morphism
550:For all
501:elements
469:Bourbaki
456:groupoid
449:groupoid
437:Clifford
433:groupoid
413:groupoid
385:groupoid
161:Lie ring
126:Semiring
21:Groupoid
3582:"Magma"
3540:0620359
3265:2371362
3133:is the
3113:colimit
3002:A000688
2996:A034382
2919:A000001
2913:A034383
2894:A057992
2873:A001426
2868:A023815
2813:A030257
2808:A076113
2789:A057991
2783:A002860
2764:A001423
2759:A023814
2740:A001425
2735:A023813
2716:A030247
2711:A090588
2692:A001329
2687:A002489
2100:, with
1595:Closure
1226:in the
1223:A001424
1202:in the
1199:A001329
1178:in the
1175:A002489
1006:strings
777:proof:
645:, then
441:Preston
292:Algebra
284:Algebra
189:Lattice
180:Lattice
3672:
3563:
3538:
3475:
3413:
3388:
3358:
3329:
3302:
3263:
3227:
2634:magma.
2488:Unital
2129:Medial
2004:Monoid
1521:Monoid
1458:, and
1376:syntax
1129:, and
807:
801:
773:, •).
401:closed
320:Graded
251:Module
242:Module
141:Domain
60:Monoid
3524:(PDF)
3497:(PDF)
3261:JSTOR
2494:Left-
2048:Group
1756:magma
1733:Magma
1531:Group
1303:) = (
898:)) •
540:magma
465:Serre
461:magma
445:Howie
381:binar
377:magma
286:-like
244:-like
182:-like
151:Field
109:-like
83:Magma
51:Group
45:-like
43:Group
3670:ISBN
3561:ISBN
3473:ISBN
3411:ISBN
3386:ISBN
3356:ISBN
3327:ISBN
3300:ISBN
3278:link
3225:ISBN
3055:has
3047:are
2358:and
2098:, •)
1891:loop
1868:Loop
1495:Loop
1332:= ((
1328:) •
1228:OEIS
1204:OEIS
1180:OEIS
1051:and
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758:, a
726:) ∗
718:) =
701:, ∗)
693:, •)
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536:, •)
439:and
375:, a
116:Ring
107:Ring
3253:doi
3151:Mag
3145:is
3143:Mag
3139:Mag
3137:of
3067:Set
3053:Mag
3037:Mag
2391:if
1340:))(
1315:)),
1307:)((
1295:• (
1279:= (
1230:).
1217:056
1214:521
1193:952
1190:981
1188:178
1169:296
1166:967
1163:294
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1038:= 2
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989:abc
975:bcd
928:≡ (
902:≡ (
890:• (
825:log
813:log
785:log
558:in
546:):
542:or
490:set
471:'s
393:set
371:In
121:Rng
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3641:.
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3620:,
3608:,
3602:,
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