1880:
1669:. The two approaches are closely related, with each having their own advantages. In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory. However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure within any of a large class of categories (namely those having finite
2868:
1380:
1786:"Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge."
1680: – an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed. Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law
772:
This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there
989:
in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements). Further, the inverse and identity are specified as morphisms in the category. For example, in a
1440:
In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study of new classes of algebras. It can enable the use of methods invented for some particular classes of
1466:
The 1956 paper by
Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially defined, typical examples for this being categories and
1691:
on the left side and omits it on the right side. At first this may seem to be a troublesome restriction, but the payoff is that operads have certain advantages: for example, one can hybridize the concepts of ring and vector space to obtain the concept of
1441:
algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it,
1471:
which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.
1761:
wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures." At the time
311:
1826:, congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by
974:
A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element
1657:. In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as
530:
Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities.
1853:
emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by
Marczewski himself, together with
1047:
over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring.
1569:
1535:
1515:
1089:
262:
242:
496:
is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all
773:
exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation
1297:). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under
1463:) were proved separately in all of these classes, but with universal algebra, they can be proven once and for all for every kind of algebraic system.
994:, the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a
222:
1766:'s algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.
2100:
Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.
264:
is an ordered sequence of natural numbers representing the arity of the operations of the algebra. However, some researchers also allow
1834:
in
Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang,
271:
1707:. Certain partial functions can also be handled by a generalization of Lawvere theories known as "essentially algebraic theories".
1831:
1358:
1091:, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.
1443:"What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."
1822:
In the period between 1935 and 1950, most papers were written along the lines suggested by
Birkhoff's papers, dealing with
338:
1913:
1401:
2384:
2296:
2276:
2216:
2197:
2178:
2149:
1757:
drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review
1427:
1791:
Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when
1409:
1487:
1481:
185:
are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like
2019:
1405:
63:
2235:
2822:
2088:
2014:. Studies in Logic and the Foundation of Mathematics. Vol. 73 (3rd ed.). North Holland. p. 1.
1495:
503:
One advantage of this restriction is that the structures studied in universal algebra can be defined in any
2670:
1908:
1974:
1362:
478:
other than equality), and in which the language used to talk about these structures uses equations only.
404:
398:
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2843:
2601:
1468:
426:
34:
themselves, not examples ("models") of algebraic structures. For instance, rather than take particular
2332:
2253:
1670:
985:
The universal algebra point of view is well adapted to category theory. For example, when defining a
510:
1934:
2848:
2838:
2547:
1390:
446:
422:
2853:
2557:
2447:
1666:
1394:
1365:
if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.
1227:
are taken off when it is clear from context which algebra the function is from.) For example, if
101:
2735:
2660:
2581:
2571:
2537:
2487:
2328:
2308:
1868:'s thesis in 1963, techniques from category theory have become important in universal algebra.
1743:
1735:
1723:
1490:. CSP refers to an important class of computational problems where, given a relational algebra
1107:
418:
135:
110:
1554:
1520:
1500:
542:. Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
2730:
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2377:
2354:
1991:
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1758:
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no quantified laws (except outermost universal quantifiers, which are allowed in varieties)
333:
After the operations have been specified, the nature of the algebra is further defined by
8:
2751:
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2621:
2607:
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2467:
2457:
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482:
436:
51:
35:
31:
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2211:, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press.
1953:
1885:
1739:
1348:
1014:
432:
1976:
The
Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
2801:
2771:
2641:
2551:
2435:
2292:
2287:
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2212:
2193:
2174:
2145:
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1879:
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442:
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2370:
2169:
2068:
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1792:
1704:
1662:
995:
616:
322:
163:
1696:, but one cannot form a similar hybrid of the concepts of group and vector space.
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2781:
2700:
2527:
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2114:
1903:
1865:
1827:
1804:
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681:: The identity element is easily seen to be unique, and is usually denoted by
168:
1796:
2886:
2720:
2685:
1893:
1854:
1830:
in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950
1812:
1677:
1640:
486:
2056:
221:). One way of talking about an algebra, then, is by referring to it as an
2704:
2637:
2611:
2492:
1898:
1843:
1763:
1711:
1648:
1299:
1095:
1040:
986:
467:
460:
2139:
2775:
2761:
2679:
2665:
2631:
2038:
George Grätzer (1968). M.H. Stone and L. Nirenberg and S.S. Chern (ed.).
1835:
1823:
1770:
1750:
1052:
999:
546:
346:
167:) is often denoted by a symbol placed between its arguments (also called
27:
1459:. Before universal algebra came along, many theorems (most notably the
2785:
2765:
2690:
2084:
1022:
769:
do, but here this is already implied by calling ∗ a binary operation.)
403:
A collection of algebraic structures defined by identities is called a
265:
1734:
had essentially the same meaning that it has today. Whitehead credits
1574:
It is proved that every computational problem can be formulated as CSP
1551:
refers to the problem whose instance is only the existential sentence
2267:
1807:
in the 1940s and 1950s furthered the field, particularly the work of
1774:
1018:
470:, typically dealing with structures having operations only (i.e. the
318:
1379:
2561:
2523:
2472:
1958:
1857:, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski,
1303:. In particular, we can take the homomorphic image of an algebra,
1026:
489:
involve an ordering relation, so would not fall within this scope.
466:
The study of equational classes can be seen as a special branch of
157:, often denoted by a symbol placed in front of its argument, like ~
2393:
2009:
1653:
Universal algebra has also been studied using the techniques of
1447:
In particular, universal algebra can be applied to the study of
500:
elements in a field, so inversion cannot be added to the type).
1952:
Zhuk, Dmitriy (2017). "The Proof of CSP Dichotomy
Conjecture".
1714:, which is sometimes described as "universal algebra + logic".
1609:
The dichotomy conjecture (proved in April 2017) states that if
1448:
306:{\displaystyle \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}
2251:
Higgins, P. J. (1963), "Algebras with a scheme of operators",
1781:, and Boole's algebra of logic. Whitehead wrote in his book:
1665:. Alternatively, one can describe algebraic structures using
1010:
Most algebraic structures are examples of universal algebras.
349:
axiom for a binary operation, which is given by the equation
334:
181:
98:
87:
2141:
An
Invitation to General Algebra and Universal Constructions
2161:
Comptes Rendus du
Premier Congrès Canadien de Mathématiques
1995:
38:
as the object of study, in universal algebra one takes the
2362:
1799:
began publishing on universal algebras. Developments in
1330:
of the sets with the operations defined coordinatewise.
1326:. A product of some set of algebraic structures is the
967:
3 equational laws (associativity, identity, and inverse)
2120:
Functorial
Semantics of Algebraic Theories (PhD Thesis)
1517:
over this algebra, the question is to find out whether
2291:, Lecture Notes in Mathematics 1533. Springer Verlag.
1486:
Universal algebra provides a natural language for the
978:, an easy exercise shows that it is unique, as is the
957:
3 operations: one binary, one unary, and one nullary (
275:
1936:
Non-dichotomies in constraint satisfaction complexity
1557:
1523:
1503:
1077:
274:
250:
230:
1875:
485:
in a wider sense fall into this scope. For example,
337:, which in universal algebra often take the form of
321:, which is an operation in the algebraic theory of
2209:Commutator Theory for Congruence Modular Varieties
2167:Burris, Stanley N., and H.P. Sankappanavar, 1981.
1563:
1529:
1509:
1475:
1083:
373:. The axiom is intended to hold for all elements
305:
256:
236:
2884:
2192:, Dordrecht, Netherlands: D. Reidel Publishing,
1676:A more recent development in category theory is
414:Restricting one's study to varieties rules out:
2163:, Toronto: University of Toronto Press: 310–326
2159:Birkhoff, Garrett (1946), "Universal algebra",
1710:Another generalization of universal algebra is
1369:
2037:
2033:
2031:
1343:, which encompass the isomorphism theorems of
2378:
2203:(First published in 1965 by Harper & Row)
1932:
1361:, which states that a class of algebras is a
129:) can be represented simply as an element of
1606:elements and a single relation, inequality.
1594:problem can be stated as CSP of the algebra
953:while the universal algebra definition has:
538:As an example, consider the definition of a
74:together with a collection of operations on
2079:Brainerd, Barron (Aug–Sep 1967) "Review of
2028:
1972:
1467:groupoids. This leads on to the subject of
1408:. Unsourced material may be challenged and
1322:that is closed under all the operations of
474:can have symbols for functions but not for
2385:
2371:
2359:—a journal dedicated to Universal Algebra.
2144:, Berkeley CA: Henry Helson, p. 398,
1742:as originators of the subject matter, and
2327:
2206:Freese, Ralph, and Ralph McKenzie, 1987.
2010:C.C. Chang and H. Jerome Keisler (1990).
1957:
1428:Learn how and when to remove this message
16:Theory of algebraic structures in general
2265:Hobby, David, and Ralph McKenzie, 1988.
2158:
1933:Bodirsky, Manuel; Grohe, Martin (2008),
1832:International Congress of Mathematicians
1231:is a constant (nullary operation), then
1051:Examples of relational algebras include
949:2 quantified laws (identity and inverse)
935:To summarize, the usual definition has:
2250:
2233:
2224:
2137:
2113:
1769:Whitehead's early work sought to unify
179:. Operations of higher or unspecified
2885:
2042:(1st ed.). Van Nostrand Co., Inc.
1333:
1066:
2366:
2318:
2285:Jipsen, Peter, and Henry Rose, 1992.
1488:constraint satisfaction problem (CSP)
1269:). If ∗ is a binary operation, then
2187:
1973:Hyland, Martin; Power, John (2007),
1951:
1406:adding citations to reliable sources
1373:
1253:. If ~ is a unary operation, then
518:is just a group in the category of
445:other than equality, in particular
425:(∀) except before an equation, and
13:
1914:Simple algebra (universal algebra)
1630:
1223:)). (Sometimes the subscripts on
1078:
251:
231:
14:
2904:
2348:
1005:
139:, often denoted by a letter like
2867:
2866:
2268:The Structure of Finite Algebras
2236:"Groups with multiple operators"
1878:
1728:A Treatise on Universal Algebra,
1378:
946:1 equational law (associativity)
121:and returns a single element of
2334:A Treatise on Universal Algebra
2271:American Mathematical Society.
2229:, D. Van Nostrand Company, Inc.
2107:
2059:A Treatise on Universal Algebra
1749:At the time structures such as
1482:Constraint satisfaction problem
1476:Constraint satisfaction problem
1131:such that, for every operation
777:and a unary operation ~, with ~
125:. Thus, a 0-ary operation (or
2341:Mainly of historical interest.
2094:
2073:
2046:
2003:
1985:
1966:
1945:
1926:
1746:with coining the term itself.
550:
1:
2170:A Course in Universal Algebra
2130:
2089:American Mathematical Monthly
1613:is a finite algebra, then CSP
45:
2671:Eigenvalues and eigenvectors
2219:. Free online second edition
1992:Essentially algebraic theory
1919:
1909:Universal algebraic geometry
1730:published in 1898, the term
1370:Motivations and applications
745:(Some authors also use the "
392:
328:
149:) is simply a function from
7:
2392:
2138:Bergman, George M. (1998),
1871:
1703:where the operators can be
1545:is often fixed, so that CSP
939:a single binary operation (
623:such that for each element
619:: There exists an element
525:
399:Variety (universal algebra)
10:
2909:
2309:General Theory of Algebras
2188:Cohn, Paul Moritz (1981),
1717:
1634:
1479:
1469:higher-dimensional algebra
689:, there exists an element
427:existential quantification
396:
85:
49:
2862:
2831:
2815:
2744:
2651:
2590:
2511:
2418:
2400:
2254:Mathematische Nachrichten
1071:We assume that the type,
533:
223:algebra of a certain type
161:. A 2-ary operation (or
143:. A 1-ary operation (or
56:In universal algebra, an
2225:Grätzer, George (1968),
1687:duplicates the variable
1564:{\displaystyle \varphi }
1530:{\displaystyle \varphi }
1510:{\displaystyle \varphi }
423:universal quantification
81:
2745:Algebraic constructions
2448:Algebraic number theory
2329:Whitehead, Alfred North
2240:Proc. London Math. Soc.
2234:Higgins, P. J. (1956),
1699:Another development is
1602:, i.e. an algebra with
1084:{\displaystyle \Omega }
1043:over a fixed field and
257:{\displaystyle \Omega }
237:{\displaystyle \Omega }
42:as an object of study.
2488:Noncommutative algebra
2319:Smith, J.D.H. (1976),
1819:, and their students.
1755:hyperbolic quaternions
1744:James Joseph Sylvester
1736:William Rowan Hamilton
1724:Alfred North Whitehead
1565:
1531:
1511:
1359:Birkhoff's HSP Theorem
1085:
565:) = (
307:
258:
238:
2725:Orthogonal complement
2498:Representation theory
2288:Varieties of Lattices
1635:Further information:
1566:
1532:
1512:
1098:between two algebras
1086:
785:. The axioms become:
308:
259:
239:
2823:Algebraic structures
2591:Algebraic structures
2576:Equivalence relation
2519:Algebraic expression
2314:Free online edition.
2281:Free online edition.
2053:Alexander Macfarlane
1759:Alexander Macfarlane
1555:
1537:can be satisfied in
1521:
1501:
1461:isomorphism theorems
1402:improve this section
1341:isomorphism theorems
1075:
713:; formally: ∀
705: =
701: =
647:; formally: ∃
639: =
635: =
577:; formally: ∀
483:algebraic structures
272:
268:operations, such as
248:
228:
32:algebraic structures
2752:Composition algebra
2676:Inner product space
2654:multilinear algebra
2542:Polynomial function
2483:Multilinear algebra
2468:Homological algebra
2458:Commutative algebra
2356:Algebra Universalis
2301:Free online edition
2182:Free online edition
2115:Lawvere, William F.
1849:In the late 1950s,
1773:(due to Hamilton),
1694:associative algebra
1494:and an existential
1334:Some basic theorems
1067:Basic constructions
904:; formally:
847:; formally:
821:Identity element:
781:usually written as
433:logical connectives
52:Algebraic structure
2532:Quadratic equation
2463:Elementary algebra
2431:Algebraic geometry
1886:Mathematics portal
1740:Augustus De Morgan
1663:algebraic theories
1661:or more generally
1561:
1527:
1507:
1144:and corresponding
1081:
877:Inverse element:
520:topological spaces
345:An example is the
303:
302:
291:
254:
234:
26:) is the field of
22:(sometimes called
2893:Universal algebra
2880:
2879:
2802:Symmetric algebra
2772:Geometric algebra
2552:Linear inequality
2503:Universal algebra
2436:Algebraic variety
2323:, Springer-Verlag
2321:Mal'cev Varieties
2227:Universal Algebra
2190:Universal Algebra
2173:Springer-Verlag.
2081:Universal Algebra
2040:Universal Algebra
1851:Edward Marczewski
1817:Andrzej Mostowski
1732:universal algebra
1705:partial functions
1587:For example, the
1580:for some algebra
1438:
1437:
1430:
1328:cartesian product
992:topological group
982:of each element.
516:topological group
514:. For example, a
323:complete lattices
276:
127:nullary operation
20:Universal algebra
2900:
2870:
2869:
2757:Exterior algebra
2426:Abstract algebra
2387:
2380:
2373:
2364:
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2338:
2324:
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2123:
2111:
2105:
2104:(1958), 731–736.
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2069:Internet Archive
2050:
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2007:
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1989:
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1982:
1981:
1970:
1964:
1963:
1961:
1949:
1943:
1942:
1941:
1930:
1888:
1883:
1882:
1809:Abraham Robinson
1793:Garrett Birkhoff
1779:Ausdehnungslehre
1686:
1659:Lawvere theories
1601:
1570:
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1567:
1562:
1536:
1534:
1533:
1528:
1516:
1514:
1513:
1508:
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1426:
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1419:
1413:
1382:
1374:
1314:A subalgebra of
1157:(of arity, say,
1122:
1090:
1088:
1087:
1082:
1061:Boolean algebras
996:closed inclusion
963:
930:
903:
893:
887:
873:
846:
837:
831:
817:
803:
789:Associativity:
685:. Then for each
617:Identity element
551:previous section
458:
409:equational class
343:equational laws.
312:
310:
309:
304:
301:
300:
290:
263:
261:
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164:binary operation
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2901:
2899:
2898:
2897:
2883:
2882:
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2876:
2858:
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2811:
2792:Quotient object
2782:Polynomial ring
2740:
2701:Linear subspace
2653:
2647:
2586:
2528:Linear equation
2507:
2453:Category theory
2414:
2396:
2391:
2351:
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2200:
2152:
2133:
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2108:
2099:
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2091:74(7): 878–880.
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1967:
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1939:
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1922:
1884:
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1874:
1866:William Lawvere
1828:Anatoly Maltsev
1805:category theory
1801:metamathematics
1720:
1701:partial algebra
1681:
1655:category theory
1651:
1645:Partial algebra
1637:Category theory
1633:
1631:Generalizations
1618:
1595:
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1318:is a subset of
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1069:
1008:
961:
905:
894:
888:
886:) =
878:
848:
838:
832:
822:
804:
802:) =
790:
679:Inverse element
536:
528:
461:order relations
450:
401:
395:
331:
317:is an infinite
296:
292:
280:
273:
270:
269:
249:
246:
245:
229:
226:
225:
220:
211:
146:unary operation
90:
84:
54:
48:
40:class of groups
24:general algebra
17:
12:
11:
5:
2906:
2896:
2895:
2878:
2877:
2875:
2874:
2863:
2860:
2859:
2857:
2856:
2851:
2846:
2844:Linear algebra
2841:
2835:
2833:
2829:
2828:
2826:
2825:
2819:
2817:
2813:
2812:
2810:
2809:
2807:Tensor algebra
2804:
2799:
2796:Quotient group
2789:
2779:
2769:
2759:
2754:
2748:
2746:
2742:
2741:
2739:
2738:
2733:
2728:
2718:
2715:Euclidean norm
2708:
2698:
2688:
2683:
2673:
2668:
2663:
2657:
2655:
2649:
2648:
2646:
2645:
2635:
2625:
2615:
2605:
2594:
2592:
2588:
2587:
2585:
2584:
2579:
2569:
2566:Multiplication
2555:
2545:
2535:
2521:
2515:
2513:
2512:Basic concepts
2509:
2508:
2506:
2505:
2500:
2495:
2490:
2485:
2480:
2478:Linear algebra
2475:
2470:
2465:
2460:
2455:
2450:
2445:
2444:
2443:
2438:
2428:
2422:
2420:
2416:
2415:
2413:
2412:
2407:
2401:
2398:
2397:
2390:
2389:
2382:
2375:
2367:
2361:
2360:
2350:
2349:External links
2347:
2345:
2344:
2325:
2316:
2306:Pigozzi, Don.
2304:
2283:
2263:
2248:
2231:
2222:
2204:
2198:
2185:
2165:
2156:
2150:
2134:
2132:
2129:
2126:
2125:
2106:
2093:
2072:
2045:
2027:
2020:
2002:
1984:
1965:
1944:
1924:
1923:
1921:
1918:
1917:
1916:
1911:
1906:
1901:
1896:
1890:
1889:
1873:
1870:
1864:Starting with
1861:, and others.
1846:, and others.
1840:Bjarni Jónsson
1789:
1788:
1719:
1716:
1632:
1629:
1614:
1575:
1560:
1546:
1541:. The algebra
1526:
1506:
1480:Main article:
1477:
1474:
1436:
1435:
1386:
1384:
1377:
1371:
1368:
1367:
1366:
1356:
1335:
1332:
1289:) ∗
1248:
1244:) =
1239:
1218:
1207:
1194:
1185:
1178:
1169:
1148:
1135:
1080:
1068:
1065:
1049:
1048:
1038:
1007:
1006:Other examples
1004:
972:
971:
968:
965:
951:
950:
947:
944:
933:
932:
875:
819:
743:
742:
676:
614:
573:) ∗
557: ∗ (
535:
532:
527:
524:
487:ordered groups
464:
463:
440:
430:
419:quantification
397:Main article:
394:
391:
369:) ∗
353: ∗ (
330:
327:
299:
295:
289:
286:
283:
279:
253:
233:
216:
209:
169:infix notation
86:Main article:
83:
80:
50:Main article:
47:
44:
15:
9:
6:
4:
3:
2:
2905:
2894:
2891:
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2800:
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2755:
2753:
2750:
2749:
2747:
2743:
2737:
2734:
2732:
2729:
2726:
2722:
2721:Orthogonality
2719:
2716:
2712:
2709:
2706:
2702:
2699:
2696:
2692:
2689:
2687:
2686:Hilbert space
2684:
2681:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2658:
2656:
2650:
2643:
2639:
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2454:
2451:
2449:
2446:
2442:
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2423:
2421:
2417:
2411:
2408:
2406:
2403:
2402:
2399:
2395:
2388:
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2381:
2376:
2374:
2369:
2368:
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2357:
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2336:
2335:
2330:
2326:
2322:
2317:
2315:
2311:
2310:
2305:
2302:
2298:
2297:0-387-56314-8
2294:
2290:
2289:
2284:
2282:
2278:
2277:0-8218-3400-2
2274:
2270:
2269:
2264:
2260:
2256:
2255:
2249:
2245:
2241:
2237:
2232:
2228:
2223:
2220:
2218:
2217:0-521-34832-3
2214:
2210:
2205:
2201:
2199:90-277-1213-1
2195:
2191:
2186:
2183:
2180:
2179:3-540-90578-2
2176:
2172:
2171:
2166:
2162:
2157:
2153:
2151:0-9655211-4-1
2147:
2143:
2142:
2136:
2135:
2122:
2121:
2116:
2110:
2103:
2097:
2090:
2086:
2082:
2076:
2070:
2067:9: 324–8 via
2066:
2062:
2060:
2054:
2049:
2041:
2034:
2032:
2023:
2017:
2013:
2006:
2000:
1998:
1993:
1988:
1978:
1977:
1969:
1960:
1955:
1948:
1938:
1937:
1929:
1925:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1894:Graph algebra
1892:
1891:
1887:
1881:
1876:
1869:
1867:
1862:
1860:
1856:
1855:Jan Mycielski
1852:
1847:
1845:
1841:
1837:
1833:
1829:
1825:
1824:free algebras
1820:
1818:
1814:
1813:Alfred Tarski
1810:
1806:
1802:
1798:
1794:
1787:
1784:
1783:
1782:
1780:
1776:
1772:
1767:
1765:
1760:
1756:
1752:
1747:
1745:
1741:
1737:
1733:
1729:
1725:
1715:
1713:
1708:
1706:
1702:
1697:
1695:
1690:
1684:
1679:
1678:operad theory
1674:
1672:
1668:
1664:
1660:
1656:
1650:
1646:
1642:
1641:Operad theory
1638:
1628:
1626:
1622:
1617:
1612:
1607:
1605:
1599:
1596:({0, 1, ...,
1593:
1591:
1585:
1583:
1578:
1572:
1558:
1549:
1544:
1540:
1524:
1504:
1497:
1493:
1489:
1483:
1473:
1470:
1464:
1462:
1458:
1454:
1450:
1445:
1444:
1432:
1429:
1421:
1411:
1407:
1403:
1397:
1396:
1392:
1387:This section
1385:
1381:
1376:
1375:
1364:
1360:
1357:
1354:
1350:
1346:
1342:
1338:
1337:
1331:
1329:
1325:
1321:
1317:
1312:
1310:
1306:
1302:
1301:
1296:
1292:
1288:
1284:
1280:
1277: ∗
1276:
1272:
1268:
1264:
1260:
1256:
1251:
1247:
1242:
1238:
1234:
1230:
1226:
1221:
1217:
1213:
1206:
1202:
1197:
1193:
1188:
1184:
1177:
1172:
1168:
1164:
1160:
1156:
1151:
1147:
1143:
1138:
1134:
1130:
1126:
1123:from the set
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1092:
1064:
1062:
1058:
1054:
1046:
1042:
1041:Vector spaces
1039:
1037:, and others.
1036:
1032:
1028:
1024:
1020:
1016:
1013:
1012:
1011:
1003:
1001:
997:
993:
988:
983:
981:
977:
969:
966:
960:
956:
955:
954:
948:
945:
942:
938:
937:
936:
929:
925:
921:
917:
913:
909:
902:
898:
892: =
891:
885:
881:
876:
872:
868:
864:
860:
856:
852:
845:
841:
836: =
835:
830: =
829:
825:
820:
816:
812:
808:
801:
797:
793:
788:
787:
786:
784:
780:
776:
770:
768:
764:
760:
756:
753: ∗
752:
749:" axiom that
748:
740:
736:
732:
728:
724:
720:
716:
712:
709: ∗
708:
704:
700:
697: ∗
696:
692:
688:
684:
680:
677:
674:
670:
666:
662:
658:
654:
650:
646:
643: ∗
642:
638:
634:
631: ∗
630:
626:
622:
618:
615:
612:
608:
604:
600:
596:
592:
588:
584:
580:
576:
572:
569: ∗
568:
564:
561: ∗
560:
556:
552:
548:
547:Associativity
545:
544:
543:
541:
531:
523:
521:
517:
513:
512:
506:
501:
499:
495:
492:The class of
490:
488:
484:
479:
477:
473:
469:
462:
457:
453:
448:
444:
441:
438:
434:
431:
428:
424:
421:, including
420:
417:
416:
415:
412:
410:
406:
400:
390:
388:
384:
380:
376:
372:
368:
365: ∗
364:
360:
357: ∗
356:
352:
348:
344:
340:
336:
326:
324:
320:
316:
297:
293:
287:
284:
281:
277:
267:
224:
219:
215:
208:
204:
200:
196:
192:
188:
184:
183:
178:
175: ∗
174:
170:
166:
165:
160:
156:
152:
148:
147:
142:
138:
137:
132:
128:
124:
120:
116:
112:
108:
104:
103:
100:
96:
89:
79:
77:
73:
70:
66:
65:
59:
53:
43:
41:
37:
33:
30:that studies
29:
25:
21:
2849:Order theory
2839:Field theory
2705:Affine space
2638:Vector space
2502:
2493:Order theory
2355:
2340:
2333:
2320:
2313:
2307:
2300:
2286:
2280:
2266:
2258:
2252:
2246:(6): 366–416
2243:
2239:
2226:
2207:
2189:
2181:
2168:
2160:
2140:
2119:
2109:
2101:
2096:
2080:
2075:
2058:
2048:
2039:
2012:Model Theory
2011:
2005:
1996:
1987:
1975:
1968:
1947:
1935:
1928:
1899:Term algebra
1863:
1848:
1844:Roger Lyndon
1821:
1790:
1785:
1768:
1764:George Boole
1751:Lie algebras
1748:
1731:
1727:
1721:
1712:model theory
1709:
1698:
1688:
1682:
1675:
1652:
1649:Model theory
1615:
1610:
1608:
1603:
1597:
1589:
1586:
1581:
1576:
1573:
1547:
1542:
1538:
1491:
1485:
1465:
1446:
1442:
1439:
1424:
1415:
1400:Please help
1388:
1323:
1319:
1315:
1313:
1308:
1304:
1300:Homomorphism
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1249:
1245:
1240:
1236:
1232:
1228:
1224:
1219:
1215:
1211:
1204:
1200:
1195:
1191:
1186:
1182:
1175:
1170:
1166:
1162:
1158:
1154:
1149:
1145:
1141:
1136:
1132:
1128:
1124:
1119:
1115:
1111:
1103:
1099:
1096:homomorphism
1093:
1070:
1053:semilattices
1050:
1009:
987:group object
984:
975:
973:
952:
934:
927:
923:
919:
915:
911:
907:
900:
896:
889:
883:
879:
870:
866:
862:
858:
854:
850:
843:
839:
833:
827:
823:
814:
810:
806:
799:
795:
791:
782:
778:
774:
771:
766:
762:
758:
754:
750:
744:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
620:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
537:
529:
508:
502:
497:
491:
480:
468:model theory
465:
455:
451:
447:inequalities
413:
408:
402:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
342:
332:
314:
217:
213:
206:
202:
198:
194:
190:
186:
180:
176:
172:
162:
158:
154:
150:
144:
140:
134:
130:
126:
122:
118:
117:elements of
114:
106:
94:
93:
91:
75:
71:
61:
57:
55:
23:
19:
18:
2854:Ring theory
2816:Topic lists
2776:Multivector
2762:Free object
2680:Dot product
2666:Determinant
2652:Linear and
2337:, Cambridge
1836:Leon Henkin
1797:Øystein Ore
1771:quaternions
1625:NP-complete
1127:to the set
1023:quasigroups
1000:cofibration
757:belongs to
549:(as in the
437:conjunction
435:other than
385:of the set
347:associative
113:that takes
28:mathematics
2832:Glossaries
2786:Polynomial
2766:Free group
2691:Linear map
2548:Inequality
2131:References
2085:P. M. Cohn
2021:0444880542
1959:1704.01914
1859:K. Urbanik
1619:is either
1418:April 2010
1261:) = ~
1190:)) =
1019:semigroups
693:such that
627:, one has
361:) = (
339:identities
266:infinitary
62:algebraic
46:Basic idea
2558:Operation
2261:: 115–132
1920:Footnotes
1775:Grassmann
1592:-coloring
1559:φ
1525:φ
1505:φ
1389:does not
1281:) =
1079:Ω
1027:groupoids
962:(2, 1, 0)
959:signature
941:signature
761:whenever
507:that has
476:relations
443:relations
393:Varieties
329:Equations
319:index set
298:α
285:∈
282:α
278:⋀
252:Ω
232:Ω
102:operation
64:structure
2887:Category
2872:Category
2582:Variable
2572:Relation
2562:Addition
2538:Function
2524:Equation
2473:K-theory
2331:(1898),
2117:(1964),
1872:See also
1726:'s book
1671:products
1496:sentence
1457:lattices
1210:), ...,
1114: :
1108:function
1057:lattices
526:Examples
511:products
505:category
498:non-zero
481:Not all
244:, where
171:), like
136:constant
111:function
2784: (
2774: (
2640: (
2630: (
2620: (
2610: (
2600: (
2410:History
2405:Outline
2394:Algebra
2065:Science
2057:Review:
2055:(1899)
1994:at the
1718:History
1600:−1}, ≠)
1449:monoids
1410:removed
1395:sources
1363:variety
1353:modules
1181:, ...,
1045:modules
980:inverse
747:closure
721:.
717: ∃
509:finite
449:, both
405:variety
133:, or a
67:) is a
58:algebra
2798:, ...)
2768:, ...)
2695:Matrix
2642:Vector
2632:theory
2622:theory
2618:Module
2612:theory
2602:theory
2441:Scheme
2295:
2275:
2215:
2196:
2177:
2148:
2018:
1667:monads
1647:, and
1455:, and
1355:, etc.
1345:groups
1059:, and
1031:magmas
534:Groups
494:fields
381:, and
335:axioms
313:where
36:groups
2736:Trace
2661:Basis
2608:Group
2598:Field
2419:Areas
2061:(pdf)
1980:(PDF)
1954:arXiv
1940:(PDF)
1904:Clone
1453:rings
1349:rings
1106:is a
1035:loops
1015:Rings
540:group
341:, or
212:,...,
201:) or
182:arity
109:is a
88:Arity
82:Arity
2731:Rank
2711:Norm
2628:Ring
2293:ISBN
2273:ISBN
2213:ISBN
2194:ISBN
2175:ISBN
2146:ISBN
2016:ISBN
1803:and
1795:and
1753:and
1738:and
1393:any
1391:cite
1339:The
1102:and
943:(2))
899:) ∗
882:∗ (~
813:) ∗
765:and
553:):
472:type
459:and
78:.
60:(or
2312:.
2087:",
2083:by
1999:Lab
1777:'s
1722:In
1685:= 1
1673:).
1623:or
1404:by
1311:).
1161:),
1153:of
1140:of
1002:).
998:(a
794:∗ (
601:)=(
439:(∧)
429:(∃)
407:or
325:.
153:to
105:on
99:ary
92:An
69:set
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