1891:
1680:. 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
2879:
1391:
1797:"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."
1691: – 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
783:
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
1000:
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
1451:
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
1477:
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
1702:
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
1452:
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,
1482:
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.
1772:
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
322:
1837:, 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
985:
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
1668:. 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
541:
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.
1864:
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
1058:
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.
1580:
1546:
1526:
1100:
273:
253:
507:
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
784:
exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation
1308:). 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
1474:) 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.
1005:, 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
233:
1777:'s algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.
2111:
Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.
275:
is an ordered sequence of natural numbers representing the arity of the operations of the algebra. However, some researchers also allow
1845:
in
Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang,
282:
1718:. Certain partial functions can also be handled by a generalization of Lawvere theories known as "essentially algebraic theories".
1842:
1369:
1102:, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.
1454:"What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."
1833:
In the period between 1935 and 1950, most papers were written along the lines suggested by
Birkhoff's papers, dealing with
349:
1924:
1412:
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2307:
2287:
2227:
2208:
2189:
2160:
1768:
drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review
1438:
17:
1802:
Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when
1420:
1498:
1492:
196:
are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like
2030:
1416:
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2099:
2025:. Studies in Logic and the Foundation of Mathematics. Vol. 73 (3rd ed.). North Holland. p. 1.
1506:
514:
One advantage of this restriction is that the structures studied in universal algebra can be defined in any
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1919:
1985:
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489:
other than equality), and in which the language used to talk about these structures uses equations only.
415:
409:
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437:
45:
themselves, not examples ("models") of algebraic structures. For instance, rather than take particular
2343:
2264:
1681:
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The universal algebra point of view is well adapted to category theory. For example, when defining a
521:
1945:
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2558:
1401:
457:
433:
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2568:
2458:
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1405:
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if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.
1238:
are taken off when it is clear from context which algebra the function is from.) For example, if
112:
2746:
2671:
2592:
2582:
2548:
2498:
2339:
2319:
1879:'s thesis in 1963, techniques from category theory have become important in universal algebra.
1754:
1746:
1734:
1501:. CSP refers to an important class of computational problems where, given a relational algebra
1118:
429:
146:
121:
1565:
1531:
1511:
553:. Usually a group is defined in terms of a single binary operation ∗, subject to the axioms:
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2002:
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no quantified laws (except outermost universal quantifiers, which are allowed in varieties)
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After the operations have been specified, the nature of the algebra is further defined by
8:
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2222:, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press.
1964:
1896:
1750:
1359:
1025:
443:
1987:
The
Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
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2782:
2652:
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79:
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2436:
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2079:
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1803:
1715:
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1006:
627:
333:
174:
1707:, but one cannot form a similar hybrid of the concepts of group and vector space.
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1914:
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692:: The identity element is easily seen to be unique, and is usually denoted by
179:
1807:
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1904:
1865:
1841:
in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950
1823:
1688:
1651:
497:
2067:
232:). One way of talking about an algebra, then, is by referring to it as an
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2648:
2622:
2503:
1909:
1854:
1774:
1722:
1659:
1310:
1106:
1051:
997:
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471:
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2642:
2049:
George Grätzer (1968). M.H. Stone and L. Nirenberg and S.S. Chern (ed.).
1846:
1834:
1781:
1761:
1063:
1010:
557:
357:
178:) is often denoted by a symbol placed between its arguments (also called
38:
1470:. Before universal algebra came along, many theorems (most notably the
2796:
2776:
2701:
2095:
1033:
780:
do, but here this is already implied by calling ∗ a binary operation.)
414:
A collection of algebraic structures defined by identities is called a
276:
1745:
had essentially the same meaning that it has today. Whitehead credits
1585:
It is proved that every computational problem can be formulated as CSP
1562:
refers to the problem whose instance is only the existential sentence
2278:
1818:
in the 1940s and 1950s furthered the field, particularly the work of
1785:
1029:
481:, typically dealing with structures having operations only (i.e. the
329:
1390:
2572:
2534:
2483:
1969:
1868:, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski,
1314:. In particular, we can take the homomorphic image of an algebra,
1037:
500:
involve an ordering relation, so would not fall within this scope.
477:
The study of equational classes can be seen as a special branch of
168:, often denoted by a symbol placed in front of its argument, like ~
2404:
2020:
1664:
Universal algebra has also been studied using the techniques of
1458:
In particular, universal algebra can be applied to the study of
511:
elements in a field, so inversion cannot be added to the type).
1963:
Zhuk, Dmitriy (2017). "The Proof of CSP Dichotomy
Conjecture".
1725:, which is sometimes described as "universal algebra + logic".
1620:
The dichotomy conjecture (proved in April 2017) states that if
1459:
317:{\displaystyle \textstyle \bigwedge _{\alpha \in J}x_{\alpha }}
2262:
Higgins, P. J. (1963), "Algebras with a scheme of operators",
1792:, and Boole's algebra of logic. Whitehead wrote in his book:
1676:. Alternatively, one can describe algebraic structures using
1021:
Most algebraic structures are examples of universal algebras.
360:
axiom for a binary operation, which is given by the equation
345:
192:
109:
98:
2152:
An
Invitation to General Algebra and Universal Constructions
2172:
Comptes Rendus du
Premier Congrès Canadien de Mathématiques
2006:
49:
as the object of study, in universal algebra one takes the
2373:
1810:
began publishing on universal algebras. Developments in
1341:
of the sets with the operations defined coordinatewise.
1337:. A product of some set of algebraic structures is the
978:
3 equational laws (associativity, identity, and inverse)
2131:
Functorial
Semantics of Algebraic Theories (PhD Thesis)
1528:
over this algebra, the question is to find out whether
2302:, Lecture Notes in Mathematics 1533. Springer Verlag.
1497:
Universal algebra provides a natural language for the
989:, an easy exercise shows that it is unique, as is the
968:
3 operations: one binary, one unary, and one nullary (
286:
1947:
Non-dichotomies in constraint satisfaction complexity
1568:
1534:
1514:
1088:
285:
261:
241:
1886:
496:
in a wider sense fall into this scope. For example,
348:, which in universal algebra often take the form of
332:, which is an operation in the algebraic theory of
2220:Commutator Theory for Congruence Modular Varieties
2178:Burris, Stanley N., and H.P. Sankappanavar, 1981.
1574:
1540:
1520:
1486:
1094:
384:. The axiom is intended to hold for all elements
316:
267:
247:
2895:
2203:, Dordrecht, Netherlands: D. Reidel Publishing,
1687:A more recent development in category theory is
425:Restricting one's study to varieties rules out:
2174:, Toronto: University of Toronto Press: 310–326
2170:Birkhoff, Garrett (1946), "Universal algebra",
1721:Another generalization of universal algebra is
1380:
2048:
2044:
2042:
1354:, which encompass the isomorphism theorems of
2389:
2214:(First published in 1965 by Harper & Row)
1943:
1372:, which states that a class of algebras is a
140:) can be represented simply as an element of
1617:elements and a single relation, inequality.
1605:problem can be stated as CSP of the algebra
964:while the universal algebra definition has:
549:As an example, consider the definition of a
85:together with a collection of operations on
2090:Brainerd, Barron (Aug–Sep 1967) "Review of
2039:
1983:
1478:groupoids. This leads on to the subject of
1419:. Unsourced material may be challenged and
1333:that is closed under all the operations of
485:can have symbols for functions but not for
2396:
2382:
2370:—a journal dedicated to Universal Algebra.
2155:, Berkeley CA: Henry Helson, p. 398,
1753:as originators of the subject matter, and
2338:
2217:Freese, Ralph, and Ralph McKenzie, 1987.
2021:C.C. Chang and H. Jerome Keisler (1990).
1968:
1439:Learn how and when to remove this message
27:Theory of algebraic structures in general
2276:Hobby, David, and Ralph McKenzie, 1988.
2169:
1944:Bodirsky, Manuel; Grohe, Martin (2008),
1843:International Congress of Mathematicians
1242:is a constant (nullary operation), then
1062:Examples of relational algebras include
960:2 quantified laws (identity and inverse)
946:To summarize, the usual definition has:
2261:
2244:
2235:
2148:
2124:
1780:Whitehead's early work sought to unify
190:. Operations of higher or unspecified
14:
2896:
2053:(1st ed.). Van Nostrand Co., Inc.
1344:
1077:
2377:
2329:
2296:Jipsen, Peter, and Henry Rose, 1992.
1499:constraint satisfaction problem (CSP)
1280:). If ∗ is a binary operation, then
2198:
1984:Hyland, Martin; Power, John (2007),
1962:
1417:adding citations to reliable sources
1384:
1264:. If ~ is a unary operation, then
529:is just a group in the category of
456:other than equality, in particular
436:(∀) except before an equation, and
24:
1925:Simple algebra (universal algebra)
1641:
1234:)). (Sometimes the subscripts on
1089:
262:
242:
25:
2915:
2359:
1016:
150:, often denoted by a letter like
2878:
2877:
2279:The Structure of Finite Algebras
2247:"Groups with multiple operators"
1889:
1739:A Treatise on Universal Algebra,
1389:
957:1 equational law (associativity)
132:and returns a single element of
2345:A Treatise on Universal Algebra
2282:American Mathematical Society.
2240:, D. Van Nostrand Company, Inc.
2118:
2070:A Treatise on Universal Algebra
1760:At the time structures such as
1493:Constraint satisfaction problem
1487:Constraint satisfaction problem
1142:such that, for every operation
788:and a unary operation ~, with ~
136:. Thus, a 0-ary operation (or
2352:Mainly of historical interest.
2105:
2084:
2057:
2014:
1996:
1977:
1956:
1937:
1757:with coining the term itself.
561:
13:
1:
2181:A Course in Universal Algebra
2141:
2100:American Mathematical Monthly
1624:is a finite algebra, then CSP
56:
2682:Eigenvalues and eigenvectors
2230:. Free online second edition
2003:Essentially algebraic theory
1930:
1920:Universal algebraic geometry
1741:published in 1898, the term
1381:Motivations and applications
756:(Some authors also use the "
403:
339:
160:) is simply a function from
7:
2403:
2149:Bergman, George M. (1998),
1882:
1714:where the operators can be
1556:is often fixed, so that CSP
950:a single binary operation (
634:such that for each element
630:: There exists an element
536:
410:Variety (universal algebra)
10:
2920:
2320:General Theory of Algebras
2199:Cohn, Paul Moritz (1981),
1728:
1645:
1490:
1480:higher-dimensional algebra
700:, there exists an element
438:existential quantification
407:
96:
60:
2873:
2842:
2826:
2755:
2662:
2601:
2522:
2429:
2411:
2265:Mathematische Nachrichten
1082:We assume that the type,
544:
234:algebra of a certain type
172:. A 2-ary operation (or
154:. A 1-ary operation (or
67:In universal algebra, an
2236:Grätzer, George (1968),
1698:duplicates the variable
1575:{\displaystyle \varphi }
1541:{\displaystyle \varphi }
1521:{\displaystyle \varphi }
434:universal quantification
92:
2756:Algebraic constructions
2459:Algebraic number theory
2340:Whitehead, Alfred North
2251:Proc. London Math. Soc.
2245:Higgins, P. J. (1956),
1710:Another development is
1613:, i.e. an algebra with
1095:{\displaystyle \Omega }
1054:over a fixed field and
268:{\displaystyle \Omega }
248:{\displaystyle \Omega }
53:as an object of study.
2499:Noncommutative algebra
2330:Smith, J.D.H. (1976),
1830:, and their students.
1766:hyperbolic quaternions
1755:James Joseph Sylvester
1747:William Rowan Hamilton
1735:Alfred North Whitehead
1576:
1542:
1522:
1370:Birkhoff's HSP Theorem
1096:
576:) = (
318:
269:
249:
2736:Orthogonal complement
2509:Representation theory
2299:Varieties of Lattices
1646:Further information:
1577:
1543:
1523:
1109:between two algebras
1097:
796:. The axioms become:
319:
270:
250:
2834:Algebraic structures
2602:Algebraic structures
2587:Equivalence relation
2530:Algebraic expression
2325:Free online edition.
2292:Free online edition.
2064:Alexander Macfarlane
1770:Alexander Macfarlane
1566:
1548:can be satisfied in
1532:
1512:
1472:isomorphism theorems
1413:improve this section
1352:isomorphism theorems
1086:
724:; formally: ∀
716: =
712: =
658:; formally: ∃
650: =
646: =
588:; formally: ∀
494:algebraic structures
283:
279:operations, such as
259:
239:
43:algebraic structures
2763:Composition algebra
2687:Inner product space
2665:multilinear algebra
2553:Polynomial function
2494:Multilinear algebra
2479:Homological algebra
2469:Commutative algebra
2367:Algebra Universalis
2312:Free online edition
2193:Free online edition
2126:Lawvere, William F.
1860:In the late 1950s,
1784:(due to Hamilton),
1705:associative algebra
1505:and an existential
1345:Some basic theorems
1078:Basic constructions
915:; formally:
858:; formally:
832:Identity element:
792:usually written as
444:logical connectives
63:Algebraic structure
2543:Quadratic equation
2474:Elementary algebra
2442:Algebraic geometry
1897:Mathematics portal
1751:Augustus De Morgan
1674:algebraic theories
1672:or more generally
1572:
1538:
1518:
1155:and corresponding
1092:
888:Inverse element:
531:topological spaces
356:An example is the
314:
313:
302:
265:
245:
37:) is the field of
33:(sometimes called
2904:Universal algebra
2891:
2890:
2813:Symmetric algebra
2783:Geometric algebra
2563:Linear inequality
2514:Universal algebra
2447:Algebraic variety
2334:, Springer-Verlag
2332:Mal'cev Varieties
2238:Universal Algebra
2201:Universal Algebra
2184:Springer-Verlag.
2092:Universal Algebra
2051:Universal Algebra
1862:Edward Marczewski
1828:Andrzej Mostowski
1743:universal algebra
1716:partial functions
1598:For example, the
1591:for some algebra
1449:
1448:
1441:
1339:cartesian product
1003:topological group
993:of each element.
527:topological group
525:. For example, a
334:complete lattices
287:
138:nullary operation
31:Universal algebra
18:Equational theory
16:(Redirected from
2911:
2881:
2880:
2768:Exterior algebra
2437:Abstract algebra
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2391:
2384:
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2213:
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2116:
2115:(1958), 731–736.
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2080:Internet Archive
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1981:
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1820:Abraham Robinson
1804:Garrett Birkhoff
1790:Ausdehnungslehre
1697:
1670:Lawvere theories
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1578:
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1527:
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1393:
1385:
1325:A subalgebra of
1168:(of arity, say,
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1101:
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1093:
1072:Boolean algebras
1007:closed inclusion
974:
941:
914:
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898:
884:
857:
848:
842:
828:
814:
800:Associativity:
696:. Then for each
628:Identity element
562:previous section
469:
420:equational class
354:equational laws.
323:
321:
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175:binary operation
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2803:Quotient object
2793:Polynomial ring
2751:
2712:Linear subspace
2664:
2658:
2597:
2539:Linear equation
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2464:Category theory
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1877:William Lawvere
1839:Anatoly Maltsev
1816:category theory
1812:metamathematics
1731:
1712:partial algebra
1692:
1666:category theory
1662:
1656:Partial algebra
1648:Category theory
1644:
1642:Generalizations
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1121:
1087:
1084:
1083:
1080:
1019:
972:
916:
905:
899:
897:) =
889:
859:
849:
843:
833:
815:
813:) =
801:
690:Inverse element
547:
539:
472:order relations
461:
412:
406:
342:
328:is an infinite
307:
303:
291:
284:
281:
280:
260:
257:
256:
240:
237:
236:
231:
222:
157:unary operation
101:
95:
65:
59:
51:class of groups
35:general algebra
28:
23:
22:
15:
12:
11:
5:
2917:
2907:
2906:
2889:
2888:
2886:
2885:
2874:
2871:
2870:
2868:
2867:
2862:
2857:
2855:Linear algebra
2852:
2846:
2844:
2840:
2839:
2837:
2836:
2830:
2828:
2824:
2823:
2821:
2820:
2818:Tensor algebra
2815:
2810:
2807:Quotient group
2800:
2790:
2780:
2770:
2765:
2759:
2757:
2753:
2752:
2750:
2749:
2744:
2739:
2729:
2726:Euclidean norm
2719:
2709:
2699:
2694:
2684:
2679:
2674:
2668:
2666:
2660:
2659:
2657:
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2636:
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2599:
2598:
2596:
2595:
2590:
2580:
2577:Multiplication
2566:
2556:
2546:
2532:
2526:
2524:
2523:Basic concepts
2520:
2519:
2517:
2516:
2511:
2506:
2501:
2496:
2491:
2489:Linear algebra
2486:
2481:
2476:
2471:
2466:
2461:
2456:
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2449:
2439:
2433:
2431:
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2409:
2408:
2401:
2400:
2393:
2386:
2378:
2372:
2371:
2361:
2360:External links
2358:
2356:
2355:
2336:
2327:
2317:Pigozzi, Don.
2315:
2294:
2274:
2259:
2242:
2233:
2215:
2209:
2196:
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2161:
2145:
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2136:
2117:
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2083:
2056:
2038:
2031:
2013:
1995:
1976:
1955:
1935:
1934:
1932:
1929:
1928:
1927:
1922:
1917:
1912:
1907:
1901:
1900:
1884:
1881:
1875:Starting with
1872:, and others.
1857:, and others.
1851:Bjarni Jónsson
1800:
1799:
1730:
1727:
1643:
1640:
1625:
1586:
1571:
1557:
1552:. The algebra
1537:
1517:
1491:Main article:
1488:
1485:
1447:
1446:
1397:
1395:
1388:
1382:
1379:
1378:
1377:
1367:
1346:
1343:
1300:) ∗
1259:
1255:) =
1250:
1229:
1218:
1205:
1196:
1189:
1180:
1159:
1146:
1091:
1079:
1076:
1060:
1059:
1049:
1018:
1017:Other examples
1015:
983:
982:
979:
976:
962:
961:
958:
955:
944:
943:
886:
830:
754:
753:
687:
625:
584:) ∗
568: ∗ (
546:
543:
538:
535:
498:ordered groups
475:
474:
451:
441:
430:quantification
408:Main article:
405:
402:
380:) ∗
364: ∗ (
341:
338:
310:
306:
300:
297:
294:
290:
264:
244:
227:
220:
180:infix notation
97:Main article:
94:
91:
61:Main article:
58:
55:
26:
9:
6:
4:
3:
2:
2916:
2905:
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2745:
2743:
2740:
2737:
2733:
2732:Orthogonality
2730:
2727:
2723:
2720:
2717:
2713:
2710:
2707:
2703:
2700:
2698:
2697:Hilbert space
2695:
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2688:
2685:
2683:
2680:
2678:
2675:
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2316:
2313:
2309:
2308:0-387-56314-8
2305:
2301:
2300:
2295:
2293:
2289:
2288:0-8218-3400-2
2285:
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2256:
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2248:
2243:
2239:
2234:
2231:
2229:
2228:0-521-34832-3
2225:
2221:
2216:
2212:
2210:90-277-1213-1
2206:
2202:
2197:
2194:
2191:
2190:3-540-90578-2
2187:
2183:
2182:
2177:
2173:
2168:
2164:
2162:0-9655211-4-1
2158:
2154:
2153:
2147:
2146:
2133:
2132:
2127:
2121:
2114:
2108:
2101:
2097:
2093:
2087:
2081:
2078:9: 324–8 via
2077:
2073:
2071:
2065:
2060:
2052:
2045:
2043:
2034:
2028:
2024:
2017:
2011:
2009:
2004:
1999:
1989:
1988:
1980:
1971:
1966:
1959:
1949:
1948:
1940:
1936:
1926:
1923:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1905:Graph algebra
1903:
1902:
1898:
1892:
1887:
1880:
1878:
1873:
1871:
1867:
1866:Jan Mycielski
1863:
1858:
1856:
1852:
1848:
1844:
1840:
1836:
1835:free algebras
1831:
1829:
1825:
1824:Alfred Tarski
1821:
1817:
1813:
1809:
1805:
1798:
1795:
1794:
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1758:
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1736:
1726:
1724:
1719:
1717:
1713:
1708:
1706:
1701:
1695:
1690:
1689:operad theory
1685:
1683:
1679:
1675:
1671:
1667:
1661:
1657:
1653:
1652:Operad theory
1649:
1639:
1637:
1633:
1628:
1623:
1618:
1616:
1610:
1607:({0, 1, ...,
1604:
1602:
1596:
1594:
1589:
1583:
1569:
1560:
1555:
1551:
1535:
1515:
1508:
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1494:
1484:
1481:
1475:
1473:
1469:
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1461:
1456:
1455:
1443:
1440:
1432:
1422:
1418:
1414:
1408:
1407:
1403:
1398:This section
1396:
1392:
1387:
1386:
1375:
1371:
1368:
1365:
1361:
1357:
1353:
1349:
1348:
1342:
1340:
1336:
1332:
1328:
1323:
1321:
1317:
1313:
1312:
1307:
1303:
1299:
1295:
1291:
1288: ∗
1287:
1283:
1279:
1275:
1271:
1267:
1262:
1258:
1253:
1249:
1245:
1241:
1237:
1232:
1228:
1224:
1217:
1213:
1208:
1204:
1199:
1195:
1188:
1183:
1179:
1175:
1171:
1167:
1162:
1158:
1154:
1149:
1145:
1141:
1137:
1134:from the set
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1103:
1075:
1073:
1069:
1065:
1057:
1053:
1052:Vector spaces
1050:
1048:, and others.
1047:
1043:
1039:
1035:
1031:
1027:
1024:
1023:
1022:
1014:
1012:
1008:
1004:
999:
994:
992:
988:
980:
977:
971:
967:
966:
965:
959:
956:
953:
949:
948:
947:
940:
936:
932:
928:
924:
920:
913:
909:
903: =
902:
896:
892:
887:
883:
879:
875:
871:
867:
863:
856:
852:
847: =
846:
841: =
840:
836:
831:
827:
823:
819:
812:
808:
804:
799:
798:
797:
795:
791:
787:
781:
779:
775:
771:
767:
764: ∗
763:
760:" axiom that
759:
751:
747:
743:
739:
735:
731:
727:
723:
720: ∗
719:
715:
711:
708: ∗
707:
703:
699:
695:
691:
688:
685:
681:
677:
673:
669:
665:
661:
657:
654: ∗
653:
649:
645:
642: ∗
641:
637:
633:
629:
626:
623:
619:
615:
611:
607:
603:
599:
595:
591:
587:
583:
580: ∗
579:
575:
572: ∗
571:
567:
563:
559:
558:Associativity
556:
555:
554:
552:
542:
534:
532:
528:
524:
523:
517:
512:
510:
506:
503:The class of
501:
499:
495:
490:
488:
484:
480:
473:
468:
464:
459:
455:
452:
449:
445:
442:
439:
435:
432:, including
431:
428:
427:
426:
423:
421:
417:
411:
401:
399:
395:
391:
387:
383:
379:
376: ∗
375:
371:
368: ∗
367:
363:
359:
355:
351:
347:
337:
335:
331:
327:
308:
304:
298:
295:
292:
288:
278:
235:
230:
226:
219:
215:
211:
207:
203:
199:
195:
194:
189:
186: ∗
185:
181:
177:
176:
171:
167:
163:
159:
158:
153:
149:
148:
143:
139:
135:
131:
127:
123:
119:
115:
114:
111:
107:
100:
90:
88:
84:
81:
77:
76:
70:
64:
54:
52:
48:
44:
41:that studies
40:
36:
32:
19:
2860:Order theory
2850:Field theory
2716:Affine space
2649:Vector space
2513:
2504:Order theory
2366:
2351:
2344:
2331:
2324:
2318:
2311:
2297:
2291:
2277:
2269:
2263:
2257:(6): 366–416
2254:
2250:
2237:
2218:
2200:
2192:
2179:
2171:
2151:
2130:
2120:
2112:
2107:
2091:
2086:
2069:
2059:
2050:
2023:Model Theory
2022:
2016:
2007:
1998:
1986:
1979:
1958:
1946:
1939:
1910:Term algebra
1874:
1859:
1855:Roger Lyndon
1832:
1801:
1796:
1779:
1775:George Boole
1762:Lie algebras
1759:
1742:
1738:
1732:
1723:model theory
1720:
1709:
1699:
1693:
1686:
1663:
1660:Model theory
1626:
1621:
1619:
1614:
1608:
1600:
1597:
1592:
1587:
1584:
1558:
1553:
1549:
1502:
1496:
1476:
1457:
1453:
1450:
1435:
1426:
1411:Please help
1399:
1334:
1330:
1326:
1324:
1319:
1315:
1311:Homomorphism
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1260:
1256:
1251:
1247:
1243:
1239:
1235:
1230:
1226:
1222:
1215:
1211:
1206:
1202:
1197:
1193:
1186:
1181:
1177:
1173:
1169:
1165:
1160:
1156:
1152:
1147:
1143:
1139:
1135:
1130:
1126:
1122:
1114:
1110:
1107:homomorphism
1104:
1081:
1064:semilattices
1061:
1020:
998:group object
995:
986:
984:
963:
945:
938:
934:
930:
926:
922:
918:
911:
907:
900:
894:
890:
881:
877:
873:
869:
865:
861:
854:
850:
844:
838:
834:
825:
821:
817:
810:
806:
802:
793:
789:
785:
782:
777:
773:
769:
765:
761:
755:
749:
745:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
693:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
621:
617:
613:
609:
605:
601:
597:
593:
589:
585:
581:
577:
573:
569:
565:
548:
540:
519:
513:
508:
502:
491:
479:model theory
476:
466:
462:
458:inequalities
424:
419:
413:
397:
393:
389:
385:
381:
377:
373:
369:
365:
361:
353:
343:
325:
228:
224:
217:
213:
209:
205:
201:
197:
191:
187:
183:
173:
169:
165:
161:
155:
151:
145:
141:
137:
133:
129:
128:elements of
125:
117:
105:
104:
102:
86:
82:
72:
68:
66:
34:
30:
29:
2865:Ring theory
2827:Topic lists
2787:Multivector
2773:Free object
2691:Dot product
2677:Determinant
2663:Linear and
2348:, Cambridge
1847:Leon Henkin
1808:Øystein Ore
1782:quaternions
1636:NP-complete
1138:to the set
1034:quasigroups
1011:cofibration
768:belongs to
560:(as in the
448:conjunction
446:other than
396:of the set
358:associative
124:that takes
39:mathematics
2843:Glossaries
2797:Polynomial
2777:Free group
2702:Linear map
2559:Inequality
2142:References
2096:P. M. Cohn
2032:0444880542
1970:1704.01914
1870:K. Urbanik
1630:is either
1429:April 2010
1272:) = ~
1201:)) =
1030:semigroups
704:such that
638:, one has
372:) = (
350:identities
277:infinitary
73:algebraic
57:Basic idea
2569:Operation
2272:: 115–132
1931:Footnotes
1786:Grassmann
1603:-coloring
1570:φ
1536:φ
1516:φ
1400:does not
1292:) =
1090:Ω
1038:groupoids
973:(2, 1, 0)
970:signature
952:signature
772:whenever
518:that has
487:relations
454:relations
404:Varieties
340:Equations
330:index set
309:α
296:∈
293:α
289:⋀
263:Ω
243:Ω
113:operation
75:structure
2898:Category
2883:Category
2593:Variable
2583:Relation
2573:Addition
2549:Function
2535:Equation
2484:K-theory
2342:(1898),
2128:(1964),
1883:See also
1737:'s book
1682:products
1507:sentence
1468:lattices
1221:), ...,
1125: :
1119:function
1068:lattices
537:Examples
522:products
516:category
509:non-zero
492:Not all
255:, where
182:), like
147:constant
122:function
2795: (
2785: (
2651: (
2641: (
2631: (
2621: (
2611: (
2421:History
2416:Outline
2405:Algebra
2076:Science
2068:Review:
2066:(1899)
2005:at the
1729:History
1611:−1}, ≠)
1460:monoids
1421:removed
1406:sources
1374:variety
1364:modules
1192:, ...,
1056:modules
991:inverse
758:closure
732:.
728: ∃
520:finite
460:, both
416:variety
144:, or a
78:) is a
69:algebra
2809:, ...)
2779:, ...)
2706:Matrix
2653:Vector
2643:theory
2633:theory
2629:Module
2623:theory
2613:theory
2452:Scheme
2306:
2286:
2226:
2207:
2188:
2159:
2029:
1678:monads
1658:, and
1466:, and
1366:, etc.
1356:groups
1070:, and
1042:magmas
545:Groups
505:fields
392:, and
346:axioms
324:where
47:groups
2747:Trace
2672:Basis
2619:Group
2609:Field
2430:Areas
2072:(pdf)
1991:(PDF)
1965:arXiv
1951:(PDF)
1915:Clone
1464:rings
1360:rings
1117:is a
1046:loops
1026:Rings
551:group
352:, or
223:,...,
212:) or
193:arity
120:is a
99:Arity
93:Arity
2742:Rank
2722:Norm
2639:Ring
2304:ISBN
2284:ISBN
2224:ISBN
2205:ISBN
2186:ISBN
2157:ISBN
2027:ISBN
1814:and
1806:and
1764:and
1749:and
1404:any
1402:cite
1350:The
1113:and
954:(2))
910:) ∗
893:∗ (~
824:) ∗
776:and
564:):
483:type
470:and
89:.
71:(or
2323:.
2098:",
2094:by
2010:Lab
1788:'s
1733:In
1696:= 1
1684:).
1634:or
1415:by
1322:).
1172:),
1164:of
1151:of
1013:).
1009:(a
805:∗ (
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450:(∧)
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