515:
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36:
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In the general case, the algebraic relations need not be associative, in which case the starting point is not the set of all words, but rather, strings punctuated with parentheses, which are used to indicate the non-associative groupings of letters. Such a string may equivalently be represented by a
2347:
1805:. Properly describing or enumerating the contents of a free object can be easy or difficult, depending on the particular algebraic object in question. For example, the free group in two generators is easily described. By contrast, little or nothing is known about the structure of
1744:
704:
1601:
2268:
2995:
1820:; one may reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable).
2944:
1769:
of strings. The identity is the empty string. In essence, the free monoid is simply the set of all words, with no equivalence relations imposed. This example is developed further in the article on the
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572:
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Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets.
1498:
159:: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include
1801:
on the leaves of the tree. Rather than starting with the collection of all possible parenthesized strings, it can be more convenient to start with the
2342:{\displaystyle {\begin{array}{ccc}S&\xrightarrow {\psi } &A\\&\searrow _{\tau }&\downarrow ^{\sigma }\\&&B\ \end{array}}}
730:
upon the words, where the relations are the defining relations of the algebraic object at hand. The free object then consists of the set of
2956:
3366:
3120:
2899:
1809:
in more than one generator. The problem of determining if two different strings belong to the same equivalence class is known as the
3315:
3237:
119:
3168:
3371:
875:; these will be given later, in the second step. Thus, one could equally well start with the alphabet in five letters that is
57:
514:
3151:
100:
532:
every map between two sets induces a unique morphism between the free objects built on them, and this defines a functor
72:
53:
3057:
3033:
2494:
2425:
723:
535:
2799:
Concretely, this sends a set into the free object on that set; it is the "inclusion of a basis". Abusing notation,
1810:
1739:{\displaystyle E=\{a_{1}a_{2}\ldots a_{n}\,\vert \;e=a_{1}a_{2}\ldots a_{n}\,;\,a_{k}\in S\,;\,n\in \mathbb {N} \}}
79:
3376:
3361:
1758:
742:
719:
2575:
3139:
3134:
699:{\displaystyle \operatorname {Hom} _{\mathbf {Set} }(X,U(B))\cong \operatorname {Hom} _{\mathbf {C} }(F(X),B).}
2354:
1077:
748:
86:
3069:
3002:
1282:
46:
2750:
2947:
2894:
199:
195:
2422:; the objects can be thought of as sets plus operations, obeying some laws. This category has a functor,
68:
2709:
2662:
1392:
1016:
3346:
3195:
3190:
3185:
1806:
878:
727:
2234:
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1356:
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507:
468:
152:
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In the next step, one imposes a set of equivalence relations. The equivalence relations for a
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430:
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175:
145:
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is the equivalence class of the identity, after the relations defining a group are imposed.
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1179:
968:, and so on, of arbitrary finite length, with the letters arranged in every possible order.
848:
818:
248:
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141:
3320:(A treatment of the one-generator free Heyting algebra is given in chapter 1, section 4.11)
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between vector spaces is entirely determined by its values on a basis of the vector space
183:
3024:
There are general existence theorems that apply; the most basic of them guarantees that
2273:
1596:{\displaystyle W(S)=\{a_{1}a_{2}\ldots a_{n}\,\vert \;a_{k}\in S\,;\,n\in \mathbb {N} \}}
93:
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The creation of free objects proceeds in two steps. For algebras that conform to the
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168:
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133:
35:
2480:. The forgetful functor is very simple: it just ignores all of the operations.
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3158:
1787:
738:
160:
3244:
2998:
815:. In the first step, there is not yet any assigned meaning to the "letters"
17:
2700:
is a function (a morphism in the category of sets), then there is a unique
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1383:
of words. Thus, in this example, the free group in two generators is the
179:
178:, in the sense that it relates to all types of algebraic structure (with
3099:
that ignores the algebra structure. It is therefore often also called a
2396:
2990:{\displaystyle \varepsilon :FU\to \operatorname {id} _{\mathbf {C} }}
2951:
1817:
2285:
1816:
As the examples suggest, free objects look like constructions from
3111:
are free symmetric and anti-symmetric algebras on a vector space.
1794:
3334:
2939:{\displaystyle \eta :\operatorname {id} _{\mathbf {Set} }\to UF}
2568:
For the free functor to be a left adjoint, one must also have a
2557:
can be thought of as the set of "generators" of the free object
718:, the first step is to consider the collection of all possible
2113:
of free generators if the following universal property holds:
1833:
29:
1071:. Applying these relations to the strings above, one obtains
745:. One starts with an alphabet consisting of the five letters
1790:; the leaves of the tree are the letters from the alphabet.
513:
232:
The following definition translates this to any category.
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2142:
2122:
2099:
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2059:
2027:
1995:
1975:
1955:
1935:
1915:
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1609:
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433:
392:
325:
299:
931:. In this example, the set of all words or strings
182:operations). It also has a formulation in terms of
60:. Unsourced material may be challenged and removed.
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2255:
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1981:
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1371:
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1005:
952:
923:
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419:
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2620:
2391:The most general setting for a free object is in
1949:. The underlying set of this algebraic structure
2526:{\displaystyle F:\mathbf {Set} \to \mathbf {C} }
2457:{\displaystyle U:\mathbf {C} \to \mathbf {Set} }
567:{\displaystyle F:\mathbf {Set} \to \mathbf {C} }
737:Consider, for example, the construction of the
257:be a concrete category with a faithful functor
186:, although this is in yet more abstract terms.
3060:, thus implying that the set of relations are
194:Free objects are the direct generalization to
8:
1793:The algebraic relations may then be general
1733:
1653:
1616:
1590:
1554:
1517:
1379:, the free object is then the collection of
1276:is the identity element. Similarly, one has
975:are that of multiplication by the identity,
918:
888:
802:
752:
574:. It follows that, if free objects exist in
1969:, often called its universe, is denoted by
27:Left adjoint to a forgetful functor to sets
2624:{\displaystyle \eta _{X}:X\to U(F(X))\,\!}
2263:such that the following diagram commutes:
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1557:
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120:Learn how and when to remove this message
3125:Specific kinds of free objects include:
2376:{\displaystyle \sigma \circ \psi =\tau }
1146:{\displaystyle aebecede=aba^{-1}b^{-1},}
3299:
808:{\displaystyle \{e,a,b,a^{-1},b^{-1}\}}
151:can be thought of as being a "generic"
3095:is the left adjoint to the functor on
2468:, which maps objects and functions in
1345:{\displaystyle abdc=abb^{-1}a^{-1}=e.}
1013:, and the multiplication of inverses:
239:is a category that is equipped with a
3310:, (1982) Cambridge University Press,
2788:{\displaystyle g=U(f)\circ \eta _{X}}
1355:Denoting the equivalence relation or
202:in a vector space. A linear function
7:
2541:to their corresponding free objects
285:of the free object to be defined. A
58:adding citations to reliable sources
3056:Here, a variety is a synonym for a
144:. Informally, a free object over a
3121:Category:Free algebraic structures
2846:) is an algebra; correctly, it is
1909:be an algebraic structure of type
293:is a pair consisting of an object
25:
3238:free partially commutative monoid
2740:{\displaystyle f\colon F(X)\to B}
2693:{\displaystyle g\colon X\to U(B)}
2639:, characterized by the following
1435:{\displaystyle F_{2}=W(S)/\sim .}
1064:{\displaystyle gg^{-1}=g^{-1}g=e}
427:, there exists a unique morphism
3169:free partially commutative group
2981:
2921:
2918:
2915:
2519:
2511:
2508:
2505:
2450:
2447:
2444:
2436:
1837:
660:
618:
615:
612:
596:; that is, there is a bijection
560:
552:
549:
546:
366:), that satisfies the following
277:be a set (that is, an object in
140:is one of the basic concepts of
34:
924:{\displaystyle S=\{a,b,c,d,e\}}
45:needs additional citations for
2972:
2927:
2877:
2874:
2868:
2862:
2856:
2834:(this abuses notation because
2821:
2815:
2809:
2769:
2763:
2731:
2728:
2722:
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2675:
2616:
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2607:
2601:
2595:
2515:
2440:
2314:
2302:
2256:{\displaystyle \sigma :A\to B}
2247:
2172:
2040:
2028:
2005:
1757:, is the monoid of all finite
1511:
1505:
1474:
1468:
1418:
1412:
947:
941:
690:
681:
675:
669:
648:
645:
639:
627:
556:
530:the universal property implies
487:
481:
443:
414:
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402:
347:
341:
335:
1:
3152:free strict monoidal category
1603:is the set of all words, and
1156:where it was understood that
960:will include strings such as
499:{\displaystyle g=U(f)\circ i}
2883:{\displaystyle X\to U(F(X))}
2181:{\displaystyle \tau :S\to B}
2073:) is a free algebra of type
2014:{\displaystyle \psi :S\to A}
1765:as alphabet, with operation
1488:{\displaystyle F_{2}=W(S)/E}
726:. Then one imposes a set of
3058:finitary algebraic category
2021:be a function. We say that
1753:. The free monoid on a set
420:{\displaystyle g:X\to U(B)}
353:{\displaystyle i:X\to U(A)}
3393:
3118:
3016:to the forgetful functor.
2635:is, up to isomorphisms in
1827:
1749:A simpler example are the
3367:Free algebraic structures
3260:free commutative semiring
3191:free distributive lattice
2827:{\displaystyle X\to F(X)}
2046:{\displaystyle (A,\psi )}
1445:This is often written as
522:If free objects exist in
506:. That is, the following
386:and any map between sets
174:The concept is a part of
3140:free commutative algebra
3135:free associative algebra
2228:, there exists a unique
592:to the faithful functor
452:{\displaystyle f:A\to B}
3233:free commutative monoid
3040:there is a free object
1824:Free universal algebras
1006:{\displaystyle ge=eg=g}
3372:Combinatorics on words
2997:, one may construct a
2991:
2940:
2895:natural transformation
2884:
2828:
2789:
2741:
2694:
2625:
2527:
2458:
2395:, where one defines a
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2343:
2257:
2222:
2202:
2182:
2150:
2130:
2107:
2087:
2067:
2047:
2015:
1983:
1963:
1943:
1923:
1903:
1883:
1740:
1597:
1489:
1436:
1373:
1346:
1270:
1250:
1249:{\displaystyle b^{-1}}
1220:
1200:
1199:{\displaystyle a^{-1}}
1170:
1147:
1065:
1007:
954:
925:
869:
868:{\displaystyle b^{-1}}
839:
838:{\displaystyle a^{-1}}
809:
700:
568:
519:
500:
453:
421:
354:
307:
3221:, and in particular,
3036:, then for every set
2992:
2941:
2885:
2829:
2790:
2742:
2695:
2626:
2528:
2459:
2378:
2344:
2258:
2223:
2203:
2183:
2151:
2149:{\displaystyle \rho }
2131:
2108:
2088:
2086:{\displaystyle \rho }
2068:
2048:
2016:
1984:
1964:
1944:
1924:
1922:{\displaystyle \rho }
1904:
1884:
1807:free Heyting algebras
1741:
1598:
1490:
1437:
1374:
1372:{\displaystyle \sim }
1347:
1271:
1251:
1221:
1201:
1171:
1148:
1066:
1008:
955:
926:
870:
840:
810:
728:equivalence relations
701:
569:
517:
501:
454:
422:
355:
308:
281:), which will be the
3306:Peter T. Johnstone,
3196:free Heyting algebra
3186:free Boolean algebra
3115:List of free objects
3097:associative algebras
2957:
2950:; together with the
2900:
2850:
2803:
2751:
2710:
2663:
2576:
2495:
2426:
2420:algebraic structures
2414:Consider a category
2355:
2269:
2235:
2212:
2192:
2160:
2140:
2120:
2097:
2077:
2057:
2053:(or informally just
2025:
1993:
1973:
1953:
1933:
1913:
1893:
1873:
1607:
1499:
1449:
1393:
1363:
1283:
1260:
1230:
1210:
1180:
1160:
1078:
1017:
979:
953:{\displaystyle W(S)}
935:
879:
849:
819:
749:
603:
536:
469:
431:
390:
323:
297:
54:improve this article
3176:free Kleene algebra
2631:. More explicitly,
2289:
2208:is the universe of
2156:and every function
1381:equivalence classes
732:equivalence classes
363:canonical injection
153:algebraic structure
3164:free abelian group
3091:construction on a
2987:
2936:
2880:
2824:
2785:
2737:
2690:
2641:universal property
2621:
2549:) in the category
2523:
2454:
2373:
2339:
2337:
2253:
2218:
2198:
2178:
2146:
2126:
2116:For every algebra
2103:
2083:
2063:
2043:
2011:
1979:
1959:
1939:
1919:
1899:
1879:
1849:. You can help by
1799:finitary relations
1736:
1593:
1485:
1432:
1369:
1342:
1266:
1246:
1226:is a stand-in for
1216:
1196:
1176:is a stand-in for
1166:
1143:
1061:
1003:
950:
921:
865:
835:
805:
696:
564:
520:
496:
449:
417:
368:universal property
350:
303:
3105:symmetric algebra
3087:For example, the
2653:is an algebra in
2483:The free functor
2466:forgetful functor
2409:forgetful functor
2334:
2290:
2221:{\displaystyle B}
2201:{\displaystyle B}
2129:{\displaystyle B}
2106:{\displaystyle S}
2066:{\displaystyle A}
1982:{\displaystyle A}
1962:{\displaystyle A}
1942:{\displaystyle S}
1902:{\displaystyle A}
1882:{\displaystyle S}
1867:
1866:
1803:Herbrand universe
1269:{\displaystyle e}
1219:{\displaystyle d}
1169:{\displaystyle c}
319:and an injection
306:{\displaystyle A}
237:concrete category
198:of the notion of
176:universal algebra
130:
129:
122:
104:
16:(Redirected from
3384:
3377:Adjoint functors
3362:Abstract algebra
3322:
3304:
3209:free Lie algebra
3109:exterior algebra
2996:
2994:
2993:
2988:
2986:
2985:
2984:
2945:
2943:
2942:
2937:
2926:
2925:
2924:
2889:
2887:
2886:
2881:
2838:is a set, while
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2825:
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2792:
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2786:
2784:
2783:
2746:
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2587:
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2529:
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2522:
2514:
2478:category of sets
2463:
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2460:
2455:
2453:
2439:
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2380:
2379:
2374:
2351:This means that
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2135:
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2110:
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2089:
2084:
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2070:
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2064:
2052:
2050:
2049:
2044:
2020:
2018:
2017:
2012:
1988:
1986:
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1980:
1968:
1966:
1965:
1960:
1948:
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1945:
1940:
1928:
1926:
1925:
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1908:
1906:
1905:
1900:
1888:
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1885:
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1862:
1859:
1841:
1834:
1745:
1743:
1742:
1737:
1732:
1710:
1709:
1695:
1694:
1682:
1681:
1672:
1671:
1651:
1650:
1638:
1637:
1628:
1627:
1602:
1600:
1599:
1594:
1589:
1567:
1566:
1552:
1551:
1539:
1538:
1529:
1528:
1494:
1492:
1491:
1486:
1481:
1461:
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1441:
1439:
1438:
1433:
1425:
1405:
1404:
1378:
1376:
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1370:
1351:
1349:
1348:
1343:
1332:
1331:
1319:
1318:
1275:
1273:
1272:
1267:
1255:
1253:
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1244:
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1223:
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1217:
1205:
1203:
1202:
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1194:
1175:
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1172:
1167:
1152:
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1149:
1144:
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1125:
1070:
1068:
1067:
1062:
1051:
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249:category of sets
241:faithful functor
231:
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142:abstract algebra
136:, the idea of a
125:
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62:
38:
30:
21:
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3297:
3285:
3202:modular lattice
3123:
3117:
3103:. Likewise the
3082:
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2393:category theory
2389:
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2336:
2335:
2324:
2323:
2313:
2311:
2301:
2297:
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2210:
2209:
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2158:
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2138:
2137:
2118:
2117:
2095:
2094:
2075:
2074:
2055:
2054:
2023:
2022:
1991:
1990:
1971:
1970:
1951:
1950:
1931:
1930:
1911:
1910:
1891:
1890:
1871:
1870:
1863:
1857:
1854:
1847:needs expansion
1832:
1826:
1779:
1701:
1686:
1673:
1663:
1642:
1629:
1619:
1605:
1604:
1558:
1543:
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1520:
1497:
1496:
1452:
1447:
1446:
1396:
1391:
1390:
1361:
1360:
1320:
1307:
1281:
1280:
1258:
1257:
1233:
1228:
1227:
1208:
1207:
1183:
1178:
1177:
1158:
1157:
1127:
1114:
1076:
1075:
1039:
1023:
1015:
1014:
977:
976:
933:
932:
877:
876:
852:
847:
846:
822:
817:
816:
789:
773:
747:
746:
722:formed from an
716:associative law
712:
654:
606:
601:
600:
593:
581:
575:
534:
533:
523:
467:
466:
460:
429:
428:
388:
387:
381:
375:
374:For any object
321:
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295:
294:
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272:
258:
252:
229:
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220:
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192:
184:category theory
165:tensor algebras
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3390:
3388:
3380:
3379:
3374:
3369:
3364:
3354:
3353:
3350:
3349:
3329:
3328:External links
3326:
3324:
3323:
3298:
3296:
3293:
3292:
3291:
3289:Generating set
3284:
3281:
3280:
3279:
3277:discrete space
3274:
3269:
3264:
3263:
3262:
3252:
3250:free semigroup
3247:
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3240:
3235:
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3206:
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3155:
3154:
3144:
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3137:
3116:
3113:
3089:tensor algebra
3081:
3078:
3068:because it is
3054:
3053:
3021:
3018:
3010:cofree functor
2983:
2978:
2974:
2971:
2968:
2965:
2962:
2946:is called the
2935:
2932:
2929:
2923:
2920:
2917:
2912:
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2905:
2879:
2876:
2873:
2870:
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2449:
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2442:
2438:
2434:
2431:
2403:, that is the
2388:
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2363:
2360:
2331:
2328:
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2102:
2082:
2062:
2042:
2039:
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2030:
2010:
2007:
2004:
2001:
1998:
1978:
1958:
1938:
1918:
1898:
1878:
1865:
1864:
1844:
1842:
1828:Main article:
1825:
1822:
1778:
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1735:
1731:
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1215:
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1033:
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987:
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862:
859:
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832:
829:
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650:
647:
644:
641:
638:
635:
632:
629:
626:
620:
617:
614:
609:
580:, the functor
562:
558:
554:
551:
548:
544:
541:
512:
511:
495:
492:
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477:
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128:
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110:September 2024
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3389:
3378:
3375:
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3365:
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3360:
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3357:
3348:
3344:
3340:
3336:
3332:
3331:
3327:
3321:
3317:
3316:0-521-23893-5
3313:
3309:
3303:
3300:
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3287:
3286:
3282:
3278:
3275:
3273:
3270:
3268:
3265:
3261:
3258:
3257:
3256:
3255:free semiring
3253:
3251:
3248:
3246:
3243:
3239:
3236:
3234:
3231:
3230:
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3174:
3170:
3167:
3165:
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3160:
3157:
3153:
3150:
3149:
3148:
3147:free category
3145:
3141:
3138:
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3133:
3132:
3131:
3128:
3127:
3126:
3122:
3114:
3112:
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3047:
3043:
3039:
3035:
3031:
3027:
3026:
3025:
3019:
3017:
3015:
3014:right adjoint
3011:
3006:
3004:
3000:
2976:
2969:
2966:
2963:
2960:
2953:
2949:
2933:
2930:
2910:
2906:
2903:
2896:
2891:
2871:
2865:
2859:
2853:
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2812:
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2780:
2776:
2772:
2766:
2760:
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2754:
2734:
2725:
2719:
2716:
2713:
2704:
2684:
2678:
2672:
2669:
2666:
2657:
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2646:
2645:
2644:
2642:
2638:
2634:
2610:
2604:
2598:
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2584:
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2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
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2490:
2486:
2481:
2479:
2475:
2471:
2467:
2432:
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2421:
2417:
2412:
2410:
2406:
2402:
2398:
2394:
2386:
2384:
2370:
2367:
2364:
2361:
2358:
2349:
2329:
2318:
2306:
2293:
2286:
2282:
2276:
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2250:
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2238:
2231:
2215:
2195:
2175:
2169:
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2143:
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2080:
2060:
2037:
2034:
2031:
2008:
2002:
1999:
1996:
1976:
1956:
1936:
1929:generated by
1916:
1896:
1889:be a set and
1876:
1861:
1852:
1848:
1845:This section
1843:
1840:
1836:
1835:
1831:
1823:
1821:
1819:
1814:
1812:
1808:
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1800:
1796:
1791:
1789:
1785:
1776:
1774:
1772:
1768:
1767:concatenation
1764:
1760:
1756:
1752:
1747:
1725:
1722:
1718:
1714:
1711:
1706:
1702:
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1664:
1660:
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1620:
1613:
1610:
1582:
1579:
1575:
1571:
1568:
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1559:
1548:
1544:
1540:
1535:
1531:
1525:
1521:
1514:
1508:
1502:
1482:
1478:
1471:
1465:
1462:
1457:
1453:
1429:
1426:
1422:
1415:
1409:
1406:
1401:
1397:
1389:
1388:
1387:
1386:
1382:
1366:
1358:
1339:
1336:
1333:
1328:
1325:
1321:
1315:
1312:
1308:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1279:
1278:
1277:
1263:
1241:
1238:
1234:
1213:
1191:
1188:
1184:
1163:
1140:
1135:
1132:
1128:
1122:
1119:
1115:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1074:
1073:
1072:
1058:
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1047:
1044:
1040:
1036:
1031:
1028:
1024:
1020:
1000:
997:
994:
991:
988:
985:
982:
974:
969:
967:
963:
944:
938:
915:
912:
909:
906:
903:
900:
897:
894:
891:
885:
882:
860:
857:
853:
830:
827:
823:
797:
794:
790:
786:
781:
778:
774:
770:
767:
764:
761:
758:
755:
744:
740:
735:
733:
729:
725:
721:
717:
709:
693:
687:
684:
678:
672:
666:
655:
651:
642:
636:
633:
630:
624:
607:
599:
598:
597:
591:
587:
584:, called the
578:
542:
539:
531:
526:
516:
509:
493:
490:
484:
478:
475:
472:
463:
446:
440:
437:
434:
411:
405:
399:
396:
393:
384:
378:
373:
372:
371:
369:
365:
364:
344:
338:
332:
329:
326:
317:
300:
288:
284:
280:
275:
269:
265:
261:
255:
250:
246:
242:
238:
233:
226:
217:
210:
206:
201:
197:
189:
187:
185:
181:
177:
172:
170:
169:free lattices
166:
162:
158:
154:
150:
147:
143:
139:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
69:"Free object"
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3347:vector space
3339:free functor
3319:
3308:Stone Spaces
3307:
3302:
3272:term algebra
3223:vector space
3181:free lattice
3130:free algebra
3124:
3101:free algebra
3093:vector space
3086:
3083:
3080:General case
3073:
3065:
3055:
3049:
3045:
3041:
3037:
3029:
3023:
3009:
3007:
2892:
2843:
2839:
2835:
2798:
2702:
2655:
2649:
2636:
2632:
2569:
2567:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2488:
2484:
2482:
2473:
2469:
2415:
2413:
2405:left adjoint
2401:free functor
2400:
2390:
2387:Free functor
2350:
2265:
2230:homomorphism
2115:
1868:
1855:
1851:adding to it
1846:
1830:Term algebra
1815:
1811:word problem
1792:
1780:
1777:General case
1762:
1754:
1751:free monoids
1748:
1444:
1354:
1155:
970:
965:
961:
736:
713:
590:left adjoint
586:free functor
585:
576:
524:
521:
461:
382:
376:
361:
360:(called the
315:
286:
282:
278:
273:
267:
263:
259:
253:
244:
234:
224:
215:
208:
204:
193:
173:
156:
148:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
18:Free functor
3343:free object
3267:free theory
3228:free monoid
3219:free module
3001:, and so a
2553:. The set
2533:takes sets
2491:. That is,
2093:on the set
1784:binary tree
1771:Kleene star
287:free object
161:free groups
138:free object
134:mathematics
3356:Categories
3214:free magma
3159:free group
3119:See also:
2747:such that
2706:-morphism
2572:-morphism
1788:free magma
1357:congruence
743:generators
739:free group
465:such that
196:categories
190:Definition
80:newspapers
3245:free ring
3066:algebraic
3028:Whenever
3020:Existence
2999:T-algebra
2973:→
2961:ε
2928:→
2904:η
2857:→
2810:→
2777:η
2773:∘
2732:→
2717::
2676:→
2670::
2647:Whenever
2596:→
2581:η
2516:→
2441:→
2371:τ
2365:ψ
2362:∘
2359:σ
2319:σ
2315:↓
2307:τ
2303:↘
2287:ψ
2248:→
2239:σ
2173:→
2164:τ
2144:ρ
2081:ρ
2038:ψ
2006:→
1997:ψ
1917:ρ
1858:June 2008
1726:∈
1712:∈
1684:…
1640:…
1583:∈
1569:∈
1541:…
1427:∼
1367:∼
1326:−
1313:−
1239:−
1189:−
1133:−
1120:−
1045:−
1029:−
858:−
828:−
795:−
779:−
667:
652:≅
625:
557:→
510:commutes:
491:∘
444:→
403:→
336:→
3283:See also
3062:finitary
2283:→
2188:, where
2136:of type
1385:quotient
1256:, while
962:aebecede
724:alphabet
710:Examples
262: :
207: :
180:finitary
3070:monadic
3034:variety
3012:is the
2407:to the
2397:functor
1795:arities
1759:strings
741:in two
508:diagram
94:scholar
3314:
3064:, and
2952:counit
2659:, and
2476:, the
2464:, the
2399:, the
2333:
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