773:. If the ring is infinite, we will thus have infinitely many operations, which is allowed by the definition of an algebraic structure in universal algebra. We will then also need infinitely many identities to express the module axioms, which is allowed by the definition of a variety of algebras. So the left
1283:
is thus enough to recover the finitary algebraic category. Indeed, finitary algebraic categories are precisely those categories equivalent to the
Eilenberg-Moore categories of finitary monads. Both these, in turn, are equivalent to categories of algebras of Lawvere theories.
1338:
is usually defined to be a class of algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. Not every author assumes that all algebras of a pseudovariety are finite; if this is the case, one sometimes talks of a
823:
proved that a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and arbitrary products. This is a result of fundamental importance to universal algebra and known as
855:
Using the easy direction of
Birkhoff's theorem, we can for example verify the claim made above, that the field axioms are not expressible by any possible set of identities: the product of fields is not a field, so fields do not form a variety.
1307:(complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of
887:
form a subvariety of the variety of semigroups because the signatures are different. Similarly, the class of semigroups that are groups is not a subvariety of the variety of semigroups. The class of monoids that are groups contains
1326:
Since varieties are closed under arbitrary direct products, all non-trivial varieties contain infinite algebras. Attempts have been made to develop a finitary analogue of the theory of varieties. This led, e.g., to the notion of
488:
847:
One direction of the equivalence mentioned above, namely that a class of algebras satisfying some set of identities must be closed under the HSP operations, follows immediately from the definitions. Proving the
1343:. For pseudovarieties, there is no general finitary counterpart to Birkhoff's theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived.
954:
920:
199:
in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation
795:
also do not form a variety of algebras, since the cancellation property is not an equation, it is an implication that is not equivalent to any set of equations. However, they do form a
741:
1404:
571:
643:
977:
do not form a subvariety, since by
Birkhoff's theorem they don't form a variety, as an arbitrary product of finitely generated abelian groups is not finitely generated.
682:
883:
Notice that although every group becomes a semigroup when the identity as a constant is omitted (and/or the inverse operation is omitted), the class of groups does
517:
forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. A sufficient defining equation is the associative law:
1396:
1185:. We may go from a variety to a finitary monad as follows. A category with some variety of algebras as objects and homomorphisms as morphisms is called a
788:
form a variety of algebras; the requirement that all non-zero elements be invertible cannot be expressed as a universally satisfied identity (see below).
356:
236:
consists of a signature, a set of variables, and a set of equational laws. Any theory gives a variety of algebras as follows. Given a theory
1177:
Besides varieties, category theorists use two other frameworks that are equivalent in terms of the kinds of algebras they describe: finitary
750:
also forms a variety of algebras. The signature here is (2,2,0,0,1) (two binary operations, two constants, and one unary operation).
1518:
1501:
974:
133:
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1328:
1315:
85:, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of
925:
891:
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to the variables in that axiom, the equation holds that is given by applying the operations to the elements of
1355:
1292:
1229:
1178:
1311:
also has infinitary operations, but their arity is countable whence its signature is small (forms a set).
792:
592:(unary). The familiar axioms of associativity, identity and inverse form one suitable set of identities:
502:
102:
1299:. This is a more general notion than "finitary algebraic category" because it admits such categories as
1331:. This kind of variety uses only finitary products. However, it uses a more general kind of identities.
811:
Given a class of algebraic structures of the same signature, we can define the notions of homomorphism,
688:
62:
1169:
etc. It has the consequence that every algebra in a variety is a homomorphic image of a free algebra.
1413:
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66:
54:
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31:
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17:
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forms a variety of algebras of signature (2,0,1), the three operations being respectively
98:
1417:
963:
is a subvariety of the variety of groups because it consists of those groups satisfying
816:
172:
94:
1529:
1509:
1433:
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Working with monads permits the following generalization. One says a category is an
960:
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1211:
1162:
796:
505:
where the objects are algebras of that theory and the morphisms are homomorphisms.
152:
148:
86:
1455:
Banaschewski, B. (1983), "The
Birkhoff Theorem for varieties of finite algebras",
844:
stand, respectively, for the operations of homomorphism, subalgebra, and product.
1166:
196:
117:
1425:
1154:
1056:
contains algebras with more than one element. One can show that for every set
812:
90:
136:. They are formally quite distinct and their theories have little in common.
1347:
514:
483:{\displaystyle f(o_{A}(a_{1},\dots ,a_{n}))=o_{B}(f(a_{1}),\dots ,f(a_{n}))}
139:
The term "variety of algebras" refers to algebras in the general sense of
1263:
799:
as the implication defining the cancellation property is an example of a
1470:
78:
1346:
Pseudovarieties are of particular importance in the study of finite
880:
and is itself a variety, i.e., is defined by a set of identities.
101:, a variety of algebras, together with its homomorphisms, forms a
30:
For the set of solutions to a system of polynomial equations, see
922:
and does not contain its subalgebra (more precisely, submonoid)
143:; there is also a more specific sense of algebra, namely as
211:
is a pair of such words; the axiom consisting of the words
1362:, describes a natural correspondence between varieties of
765:. To express the scalar multiplication with elements from
203:
has as many branches away from the root as the arity of
1513:, Lecture Notes in Mathematics 1533. Springer Verlag.
167:(in this context) is a set, whose elements are called
928:
894:
691:
652:
601:
526:
359:
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A variety of algebras should not be confused with an
82:
1405:Proceedings of the Cambridge Philosophical Society
948:
914:
769:, we need one unary operation for each element of
735:
676:
637:
565:
482:
1490:Stanley N. Burris and H.P. Sankappanavar (1981),
1072:. This means that there is an injective set map
949:{\displaystyle \langle \mathbb {N} ,+\rangle }
915:{\displaystyle \langle \mathbb {Z} ,+\rangle }
8:
943:
929:
909:
895:
1052:is a non-trivial variety of algebras, i.e.
1366:and pseudovarieties of finite semigroups.
309:. The class of algebras of a given theory
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777:-modules do form a variety of algebras.
1397:"On the structure of abstract algebras"
1387:
1314:Every finitary algebraic category is a
1303:(complete atomic Boolean algebras) and
1189:. For any finitary algebraic category
1486:Two monographs available free online:
132:, which means a set of solutions to a
69:form a variety of algebras, as do the
7:
1507:Peter Jipsen and Henry Rose (1992),
1262:, meaning it commutes with filtered
973:, with no change of signature. The
301:as indicated by the trees defining
293:and each assignment of elements of
736:{\displaystyle xx^{-1}=x^{-1}x=1.}
252:together with, for each operation
25:
975:finitely generated abelian groups
1322:Pseudovariety of finite algebras
1153:This generalizes the notions of
1004:, meaning that for any objects
876:that has the same signature as
806:
757:, we can consider the class of
320:Given two algebras of a theory
1493:A Course in Universal Algebra.
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1089:that satisfies the following
107:finitary algebraic categories
27:Class of algebraic structures
1329:variety of finite semigroups
1316:locally presentable category
1233:, meaning that the category
566:{\displaystyle x(yz)=(xy)z.}
187:, whose elements are called
1358:, often referred to as the
1187:finitary algebraic category
984:and its homomorphisms as a
638:{\displaystyle x(yz)=(xy)z}
105:; these are usually called
18:Birkhoff's HSP theorem
1552:
1341:variety of finite algebras
826:Birkhoff's variety theorem
807:Birkhoff's Variety theorem
753:If we fix a specific ring
588:(nullary, a constant) and
175:(0, 1, 2, ...) called its
61:satisfying a given set of
29:
1426:10.1017/S0305004100013463
1395:Birkhoff, G. (Oct 1935),
1016:, the homomorphisms from
868:of a variety of algebras
283:such that for each axiom
1239:Eilenberg–Moore category
1115:, there exists a unique
1028:are exactly those from
793:cancellative semigroups
677:{\displaystyle 1x=x1=x}
1352:formal language theory
1254:. Moreover the monad
959:However, the class of
950:
916:
737:
678:
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120:of a given signature.
118:coalgebraic structures
1510:Varieties of Lattices
1237:is equivalent to the
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501:. Any theory gives a
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179:. Given a signature
1093:: given any algebra
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493:for every operation
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145:algebra over a field
116:is the class of all
97:. In the context of
55:algebraic structures
1458:Algebra Universalis
1418:1935PCPS...31..433B
1356:Eilenberg's theorem
315:variety of algebras
65:. For example, the
43:variety of algebras
1471:10.1007/BF01194543
1289:algebraic category
1159:free abelian group
1091:universal property
980:Viewing a variety
946:
912:
733:
674:
635:
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248:consists of a set
83:Birkhoff's theorem
81:etc. According to
1536:Universal algebra
1496:Springer-Verlag.
1364:regular languages
1195:forgetful functor
872:is a subclass of
513:The class of all
141:universal algebra
130:algebraic variety
95:(direct) products
39:universal algebra
32:Algebraic variety
16:(Redirected from
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1309:sigma algebras
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1066:free algebra F
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1438:the original
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1376:Quasivariety
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1163:free algebra
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860:Subvarieties
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149:vector space
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1167:free module
1064:contains a
830:HSP theorem
260:with arity
197:rooted tree
124:Terminology
91:subalgebras
87:homomorphic
57:of a given
1348:semigroups
1269:The monad
1155:free group
1136:such that
866:subvariety
828:or as the
813:subalgebra
584:(binary),
515:semigroups
350:such that
183:and a set
169:operations
159:Definition
63:identities
1434:121173630
1291:if it is
944:⟩
930:⟨
910:⟩
896:⟨
717:−
701:−
590:inversion
497:of arity
453:…
393:…
189:variables
165:signature
147:, i.e. a
114:covariety
59:signature
1530:Category
1370:See also
1274: :
1264:colimits
1260:finitary
1218: :
1201: :
1124: :
1106: :
1077: :
1048:Suppose
986:category
850:converse
763:-modules
586:identity
509:Examples
503:category
341: :
274: :
153:bilinear
103:category
89:images,
1414:Bibcode
1293:monadic
1230:monadic
817:product
242:algebra
79:monoids
53:of all
49:is the
1517:
1500:
1432:
1210:has a
1193:, the
1179:monads
840:, and
815:, and
782:fields
578:groups
324:, say
234:theory
207:. An
181:σ
93:, and
77:, the
73:, the
67:groups
1453:E.g.
1441:(PDF)
1430:S2CID
1400:(PDF)
1382:Notes
1305:CSLat
1295:over
996:is a
759:left
748:rings
240:, an
177:arity
75:rings
51:class
1515:ISBN
1498:ISBN
1301:CABA
1181:and
1070:on S
791:The
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