3414:
386:
5005:
5492:
5290:
5049:
is being sharded. To finalize reduction properly, the "Shuffling" stage regroups the data among the nodes. If we do not need this step, the whole Map/Reduce consists of mapping and reducing; both operations are parallelizable, the former due to its element-wise nature, the latter due to associativity
4795:" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be
3063:
The cancellative property in a monoid is not necessary to perform the
Grothendieck construction β commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, if
4832:
4754:
In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.
2260:
5301:
3059:
The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group.
2379:
5038:). MapReduce, in computing, consists of two or three operations. Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. "Reduce" consists of folding those elements, so that in the end we produce just one element.
5108:
5804:
5000:{\displaystyle \mathrm {fold} :M^{*}\rightarrow M=\ell \mapsto {\begin{cases}\varepsilon &{\mbox{if }}\ell =\mathrm {nil} \\m\bullet \mathrm {fold} \,\ell '&{\mbox{if }}\ell =\mathrm {cons} \,m\,\ell '\end{cases}}}
2154:
3608:
between monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of the homomorphism. For example, consider
4066:
2544:
3691:, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted.
2265:
3974:
5487:{\displaystyle \sum _{j\in J}{\sum _{i\in I_{j}}{m_{i}}}=\sum _{i\in I}m_{i}\quad {\text{ if }}\bigcup _{j\in J}I_{j}=I{\text{ and }}I_{j}\cap I_{j'}=\emptyset \quad {\text{ for }}j\neq j'}
5045:, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case, the
5652:
4133:
5091:
5285:{\displaystyle \sum _{i\in \emptyset }{m_{i}}=0;\quad \sum _{i\in \{j\}}{m_{i}}=m_{j};\quad \sum _{i\in \{j,k\}}{m_{i}}=m_{j}+m_{k}\quad {\text{ for }}j\neq k}
5724:
372:
2923:
is not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse of
3698:, as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only element
5934:
Some authors omit the requirement that a submonoid must contain the identity element from its definition, requiring only that it have
5035:
3989:
2255:{\displaystyle {\begin{bmatrix}0&1&2&\cdots &n-2&n-1\\1&2&3&\cdots &n-1&k\end{bmatrix}}}
6500:
Automata, Languages and
Programming: 17th International Colloquium, Warwick University, England, July 16β20, 1990, Proceedings
6534:
6459:
6358:
2486:
714:
with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a
3930:
365:
17:
6470:
Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers
1176:
6610:
6576:
6511:
6477:
6438:
6409:
6301:
2891:
Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements
6523:
Algebraic foundations in computer science. Essays dedicated to Symeon
Bozapalidis on the occasion of his retirement
6376:
Hebisch, Udo (1992). "Eine algebraische
Theorie unendlicher Summen mit Anwendungen auf Halbgruppen und Halbringe".
6145:
740:
Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by
5014:
can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a
5834:
3049:
and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group.
2374:{\displaystyle f(i):={\begin{cases}i+1,&{\text{if }}0\leq i<n-1\\k,&{\text{if }}i=n-1.\end{cases}}}
1329:- monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices,
6726:
6680:
358:
4693:
composition when restricted to the set of all morphisms whose source and target is a given object. That is,
450:
The functions from a set into itself form a monoid with respect to function composition. More generally, in
4792:
513:
1509:, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity
6675:
1334:
227:
5607:
4082:
1179:, four have a two-sided identity that is also commutative and associative. These four each make the set
6560:
3378:
1870:
458:
to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
6736:
4729:
1564:
470:
6329:
4883:
2289:
6731:
4766:
3738:
3402:
1838:
455:
31:
4711:, one can construct a small category with only one object and whose morphisms are the elements of
3045:) is constructed from the additive monoid of natural numbers (a commutative monoid with operation
1683:, with the binary operation and identity element defined on corresponding coordinates, called the
6293:
5069:
521:
318:
6495:
1020:). Commutative monoids are often written additively. Any commutative monoid is endowed with its
6324:
4258:
3605:
1914:
1163:
715:
525:
38:
3037:
A commutative monoid with the cancellation property can always be embedded in a group via the
6404:, London Mathematical Society Monographs. New Series, vol. 12, Oxford: Clarendon Press,
6397:
6155:
4202:
4187:
2951:
1555:
1491:
1196:
1188:
1127:
933:
is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the
917:
474:
6670:
3413:
6586:
5829:
4207:
4192:
1861:
1572:
1547:
1460:
722:-ary (or nullary) operation. The monoid therefore is characterized by specification of the
506:
490:
466:
305:
297:
269:
264:
255:
212:
154:
6654:
6594:
6544:
6487:
6419:
6389:
6368:
6311:
8:
6521:
Kuich, Werner (2011). "Algebraic systems and pushdown automata". In Kuich, Werner (ed.).
5848:
4784:
4733:
4640:
2874:
1446:
1365:
1200:
1184:
934:
786:
529:
444:
394:
323:
313:
164:
64:
56:
47:
385:
6642:
4744:
4618:
3748:
3729:. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.
3695:
3189:
2886:
2873:
The set of all invertible elements in a monoid, together with the operation β’, forms a
1964:
129:
120:
78:
4769:
which is an abstract definition of what is a monoid in a category. A monoid object in
6689:
6606:
6572:
6530:
6507:
6473:
6455:
6434:
6405:
6354:
6297:
5598:
3394:
3105:
1865:
1721:
1671:
548:
482:
6646:
4803:
or similar algorithm, in order to utilize multiple cores or processors efficiently.
1158:
A monoid for which the operation is commutative for some, but not all elements is a
926:
is not always a submonoid, since the identity elements may differ. For example, the
6650:
6632:
6590:
6564:
6540:
6483:
6415:
6385:
6364:
6349:. Operations Research/Computer Science Interfaces Series. Vol. 41. Dordrecht:
6334:
6307:
5824:
5509:
4770:
4214:
4197:
1250:
1243:
707:
557:
486:
462:
425:
421:
406:
149:
6472:, de Gruyter Expositions in Mathematics, vol. 29, Berlin: Walter de Gruyter,
5799:{\displaystyle \sum _{I}a_{i}=\sup _{{\text{finite }}E\subset I}\;\sum _{E}a_{i},}
4689:. Indeed, the axioms required of a monoid operation are exactly those required of
174:
6620:
6582:
6559:, Encyclopedia of Mathematics and Its Applications, vol. 17 (2nd ed.),
6526:
6503:
6447:
6426:
6350:
4796:
4686:
4348:
4325:
3980:
3763:
3389:
2703:
1968:
1551:
1521:
1330:
1318:
782:
711:
517:
502:
451:
390:
241:
235:
222:
202:
193:
159:
96:
6338:
5843:
5019:
5011:
4483:
4460:
4072:
1684:
1529:
1506:
1215:
920:
under the monoid operation, and is a monoid for this inherited operation, then
498:
283:
3405:
can be made into an operator monoid by adjoining the identity transformation.
539:
for the history of the subject, and some other general properties of monoids.
6720:
6637:
6568:
6160:
4758:
Monoids, just like other algebraic structures, also form their own category,
4662:
3618:
3387:-acts are defined in a similar way. A monoid with an act is also known as an
3109:
1972:
1961:
1957:. For more on the relationship between category theory and monoids see below.
1654:
endowed with monoid structure (or, in general, any finite number of monoids,
1369:
1358:
1343:
1326:
1322:
1130:
1123:
927:
741:
169:
134:
91:
6692:
4698:
A monoid is, essentially, the same thing as a category with a single object.
3770:
to monoid congruences, and then constructing the quotient monoid, as above.
829:
that is closed under the monoid operation and contains the identity element
6285:
5853:
5594:
4438:
4415:
4236:
4079:(it has infinite order). Elements of this plactic monoid may be written as
3449:
3398:
2629:
As a special case, one can define nonnegative integer powers of an element
2420:
1930:
1576:
1429:
1159:
494:
343:
274:
108:
4762:, whose objects are monoids and whose morphisms are monoid homomorphisms.
1975:. Its unit element is the class of the ordinary 2-sphere. Furthermore, if
6552:
5601:, and these least upper bounds are compatible with the monoid operation:
5015:
3756:
3726:
3218:: equivalently, that no element other than zero has an additive inverse.
3041:. That is how the additive group of the integers (a group with operation
1585:
1525:
1307:
1009:
478:
418:
410:
333:
328:
217:
207:
181:
27:
Algebraic structure with an associative operation and an identity element
6710:
5858:
4800:
4574:
4551:
4393:
4370:
3227:
1311:
1231:
1225:
1208:
83:
6697:
5094:
5031:
4717:. The composition of morphisms is given by the monoid operation
4528:
4505:
3722:
1754:
1731:
1376:
1300:
703:
536:
440:
398:
338:
144:
101:
69:
778:. This notation does not imply that it is numbers being multiplied.
5839:
5063:
5046:
5042:
4788:
4690:
4303:
4280:
1954:
1024:
139:
6319:
Droste, M.; Kuich, W (2009), "Semirings and Formal Power Series",
5815:
together with this infinitary sum operation is a complete monoid.
3747:, much in the same way that groups can be specified by means of a
6496:"Ο-continuous semirings, algebraic systems and pushdown automata"
4728:
between single object categories. So this construction gives an
4725:
3694:
In contrast, a semigroup homomorphism between groups is always a
1517:
429:
6706:
1432:
between the category of semigroups and the category of monoids.
1428:. This conversion of any semigroup to the monoid is done by the
1278:
is a commutative monoid under intersection (identity element is
5030:
An application of monoids in computer science is the so-called
4787:
can be endowed with a monoid structure. In a common pattern, a
3597:
respectively. Monoid homomorphisms are sometimes simply called
1768:
1259:
is a commutative monoid under multiplication (identity element
815:
6605:, Springer Monographs in Mathematics, vol. 71, Springer,
1991:
denotes the class of the projective plane, then every element
6525:. Lecture Notes in Computer Science. Vol. 7020. Berlin:
4739:
and a full subcategory of the category of (small) categories
3870:. Finally, one takes the reflexive and transitive closure of
1982:
1299:
is a commutative monoid under union (identity element is the
723:
4061:{\displaystyle \langle a,b\,\vert \;aba=baa,bba=bab\rangle }
3184:. If an inverse monoid is cancellative, then it is a group.
6468:
Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander V. (2000),
6224:
4993:
2367:
1384:
may be turned into a monoid simply by adjoining an element
6133:
3284:
which is compatible with the monoid structure as follows:
1639:. Any commutative monoid is the opposite monoid of itself.
1357:
forms the trivial (one-element) monoid, which is also the
6184:
1224:
is a commutative monoid under addition (identity element
2132:, and every cyclic monoid is isomorphic to one of these.
1777:. A binary operation for such subsets can be defined by
432:
with addition form a monoid, the identity element being
6347:
Graphs, Dioids and
Semirings: New Models and Algorithms
6214:
6212:
6210:
3094:
1579:
serves as the identity element. This monoid is denoted
4951:
4892:
2539:{\displaystyle p_{n}=\textstyle \prod _{i=1}^{n}a_{i}}
2503:
2163:
1183:
a commutative monoid. Under the standard definitions,
901:
is a monoid under the binary operation inherited from
6172:
5938:
identity element, which can be distinct from that of
5727:
5610:
5304:
5111:
5072:
4835:
4085:
3992:
3933:
2489:
2268:
2157:
1494:
of a sequence given the orders of its elements, with
6502:. Lecture Notes in Computer Science. Vol. 443.
6263:
6261:
6207:
6236:
6196:
6122:
5018:might differ depending on pre-order vs. post-order
6621:"Tensor products of structures with interpolation"
6467:
5798:
5646:
5486:
5284:
5085:
4999:
4127:
4060:
3968:
3751:. One does this by specifying a set of generators
3377:This is the analogue in monoid theory of a (left)
3238:be a monoid, with the binary operation denoted by
2538:
2437:The monoid axioms imply that the identity element
2373:
2254:
1997:of the monoid has a unique expression in the form
6603:The q-theory of Finite Semigroups: A New Approach
6258:
6247:
5036:Encoding Map-Reduce As A Monoid With Left Folding
3969:{\displaystyle \langle p,q\,\vert \;pq=1\rangle }
3052:If a monoid has the cancellation property and is
1615:has the same carrier set and identity element as
1439:) may be formed by adjoining an identity element
1306:Generalizing the previous example, every bounded
6718:
5752:
5626:
5617:
1162:; trace monoids commonly occur in the theory of
744:; for example, the monoid axioms may be written
6687:
6600:
6151:
5690:is a continuous monoid, then for any index set
3885:is simply given as a set of equations, so that
3790:. This can be extended to a symmetric relation
1714:. The set of all functions from a given set to
1532:, with addition or multiplication as operation.
1435:Thus, an idempotent monoid (sometimes known as
1153:
4778:
4174:
2140:can be considered as a function on the points
6344:
6139:
4747:is equivalent to another full subcategory of
3627:equipped with multiplication. In particular,
1478:to the left-zero semigroup with two elements
366:
5224:
5212:
5167:
5161:
4685:Monoids can be viewed as a special class of
4055:
4006:
3993:
3963:
3947:
3934:
3221:
1720:is also a monoid. The identity element is a
6318:
6230:
5959:in the monoid. Since the monoid is finite,
1831:. In the same way the power set of a group
447:occur in several branches of mathematics.
30:For monoid objects in category theory, see
6601:Rhodes, John; Steinberg, Benjamin (2009),
5772:
4009:
3950:
3732:
2752:. Inverses, if they exist, are unique: if
984:. If there is a finite set that generates
581:if it satisfies the following two axioms:
516:, the study of monoids is fundamental for
373:
359:
6636:
6328:
5980:. But then, by cancellation we have that
5062:is a commutative monoid equipped with an
4981:
4977:
4939:
4005:
3946:
3193:is an additively written monoid in which
2819:, then one can define negative powers of
1337:are endowed with these monoid structures.
6551:
6446:
6433:, vol. I, D. Van Nostrand Company,
6425:
6345:Gondran, Michel; Minoux, Michel (2008).
6190:
6178:
6166:
6128:
4724:Likewise, monoid homomorphisms are just
3412:
2455:are identity elements of a monoid, then
1455:. The opposite monoid (sometimes called
384:
6618:
6375:
6242:
6202:
3979:is the equational presentation for the
3879:In the typical situation, the relation
3780:, one defines its symmetric closure as
3725:monoid homomorphism is called a monoid
3408:
2395:is then given by function composition.
1907:Generalizing the previous example, let
1730:; the associative operation is defined
1482:. Then the resulting idempotent monoid
1207:. The monoids from AND and OR are also
14:
6719:
6284:
6218:
6169:, p. 29, examples 1, 2, 4 & 5
3762:. One does this by extending (finite)
3444:. It is injective, but not surjective.
3104:. In particular, if the monoid has an
2691:
2572:elements of a monoid recursively: let
2472:
1953:, forms a monoid under composition of
1230:) or multiplication (identity element
1211:while those from XOR and XNOR are not.
781:A monoid in which each element has an
6688:
6520:
6493:
6454:, vol. 1 (2nd ed.), Dover,
6396:
6292:. Oxford Logic Guides. Vol. 49.
6267:
6253:
6102:is the identity of its domain monoid
3876:, which is then a monoid congruence.
3108:, then its Grothendieck group is the
2880:
1892:elements, the monoid of functions on
1724:mapping any value to the identity of
1003:
3242:and the identity element denoted by
1823:into a monoid with identity element
792:
5053:
4806:Given a sequence of values of type
3659:is a semigroup homomorphism, since
3115:
1850:be a set. The set of all functions
24:
6378:Bayreuther Mathematische Schriften
5895:satisfy the above equations, then
5647:{\displaystyle a+\sup S=\sup(a+S)}
5461:
5123:
5074:
4973:
4970:
4967:
4964:
4935:
4932:
4929:
4926:
4911:
4908:
4905:
4846:
4843:
4840:
4837:
4128:{\displaystyle a^{i}b^{j}(ba)^{k}}
3635:is the identity element. Function
1687:, is also a monoid (respectively,
1621:, and its operation is defined by
25:
6748:
6663:
6498:. In Paterson, Michael S. (ed.).
5585:is an ordered commutative monoid
4826:operation is defined as follows:
2126:gives a distinct monoid of order
710:. It can also be thought of as a
6402:Fundamentals of Semigroup Theory
6091:is a semigroup homomorphism and
3755:, and a set of relations on the
2954:(or is cancellative) if for all
2813:is invertible, say with inverse
1353:closed under a binary operation
916:is a subset of a monoid that is
6019:
5464:
5387:
5267:
5200:
5149:
4818:and associative operation
4703:More precisely, given a monoid
3039:Grothendieck group construction
428:. For example, the nonnegative
6625:Pacific Journal of Mathematics
5947:
5928:
5871:
5835:Monad (functional programming)
5641:
5629:
4875:
4863:
4116:
4106:
3056:, then it is in fact a group.
2383:Multiplication of elements in
2278:
2272:
702:In other words, a monoid is a
13:
1:
6321:Handbook of Weighted Automata
6277:
4153:, as the relations show that
3393:. Important examples include
3187:In the opposite direction, a
2483:, one can define the product
2477:For each nonnegative integer
2432:
1116:. This is often used in case
966:if the smallest submonoid of
940:
797:
542:
389:Algebraic structures between
6431:Lectures in Abstract Algebra
6115:
5996:is the identity. Therefore,
5025:
4791:of elements of a monoid is "
4573:Commutative-and-associative
3417:Example monoid homomorphism
3124:is a monoid where for every
2058:be a cyclic monoid of order
1154:Partially commutative monoid
1138:, in which case we say that
1008:A monoid whose operation is
658:such that for every element
514:theoretical computer science
7:
6676:Encyclopedia of Mathematics
6619:Wehrung, Friedrich (1996).
6339:10.1007/978-3-642-01492-5_1
6152:Rhodes & Steinberg 2009
5818:
5696:and collection of elements
5086:{\displaystyle \Sigma _{I}}
4779:Monoids in computer science
4734:category of (small) monoids
4175:Relation to category theory
3269:together with an operation
2706:if there exists an element
1864:. The identity is just the
1593:. It is not commutative if
1321:can be endowed with both a
1317:In particular, any bounded
1242:under addition is called a
1169:
397:. For example, monoids are
10:
6753:
6561:Cambridge University Press
5500:ordered commutative monoid
5041:For example, if we have a
4783:In computer science, many
4765:There is also a notion of
3736:
3225:
2884:
2842:; this makes the equation
2017:is a positive integer and
1871:full transformation monoid
1597:has at least two elements.
1505:The underlying set of any
1082:such that for any element
473:built from a given set of
417:is a set equipped with an
36:
29:
6140:Gondran & Minoux 2008
3222:Acts and operator monoids
2746:is called the inverse of
1981:denotes the class of the
1567:over some fixed alphabet
998:finitely generated monoid
501:provide a foundation for
6638:10.2140/pjm.1996.176.267
6569:10.1017/CBO9780511566097
5864:
5502:is a commutative monoid
3773:Given a binary relation
3739:Presentation of a monoid
3403:transformation semigroup
3136:, there exists a unique
2770:, then by associativity
1868:. It is also called the
1839:product of group subsets
1550:over a given ring, with
1293:, the set of subsets of
1272:, the set of subsets of
1177:binary Boolean operators
1100:in the set generated by
1064:of a commutative monoid
646:There exists an element
37:Not to be confused with
32:Monoid (category theory)
6294:Oxford University Press
6231:Droste & Kuich 2009
5034:programming model (see
3743:Monoids may be given a
3733:Equational presentation
2419:, and gives the unique
1175:Out of the 16 possible
1016:(or, less commonly, an
573:, which we will denote
489:are used in describing
6557:Combinatorics on words
6494:Kuich, Werner (1990).
5953:Proof: Fix an element
5800:
5648:
5488:
5286:
5087:
5001:
4812:with identity element
4180:Group-like structures
4129:
4062:
3970:
3606:semigroup homomorphism
3585:are the identities on
3445:
2939:, which is not true.
2540:
2524:
2375:
2256:
1837:is a monoid under the
1575:as the operation. The
1563:The set of all finite
1498:representing equality.
1368:is a monoid and every
1164:concurrent computation
910:On the other hand, if
526:formal language theory
454:, the morphisms of an
402:
5801:
5649:
5489:
5287:
5088:
5002:
4130:
4063:
3971:
3924:. Thus, for example,
3416:
2952:cancellation property
2541:
2504:
2376:
2257:
1860:forms a monoid under
1747:and identity element
1556:matrix multiplication
1492:lexicographical order
1459:) is formed from the
1372:a commutative monoid.
491:finite-state machines
443:with identity. Such
388:
6727:Algebraic structures
6529:. pp. 228β256.
6506:. pp. 103β110.
5725:
5667:and directed subset
5608:
5302:
5109:
5070:
4833:
4083:
3990:
3931:
3452:between two monoids
3409:Monoid homomorphisms
2487:
2411:is a permutation of
2266:
2155:
1862:function composition
1573:string concatenation
1571:forms a monoid with
1461:right zero semigroup
1314:commutative monoid.
1144:is an order-unit of
935:nonnegative integers
507:concurrent computing
467:computer programming
445:algebraic structures
270:Group with operators
213:Complemented lattice
48:Algebraic structures
5849:Star height problem
5718:, one can define
4785:abstract data types
4181:
4159:commutes with both
3006:, and the equality
2692:Invertible elements
2473:Products and powers
1753:, and consider its
1743:with the operation
1472:Adjoin an identity
1447:left zero semigroup
530:star height problem
522:KrohnβRhodes theory
324:Composition algebra
84:Quasigroup and loop
6690:Weisstein, Eric W.
6193:, p. 31, Β§1.2
5796:
5782:
5771:
5737:
5644:
5484:
5408:
5376:
5344:
5320:
5282:
5228:
5171:
5127:
5083:
4997:
4992:
4955:
4896:
4775:is just a monoid.
4745:category of groups
4619:Commutative monoid
4179:
4125:
4058:
3966:
3749:group presentation
3696:group homomorphism
3446:
3395:transition systems
3257:(or left act over
3190:zerosumfree monoid
2917:holds even though
2887:Grothendieck group
2881:Grothendieck group
2536:
2535:
2405:then the function
2371:
2366:
2252:
2246:
1767:consisting of all
1583:and is called the
1236:). A submonoid of
1203:have the identity
1191:have the identity
1014:commutative monoid
1004:Commutative monoid
847:is a submonoid of
483:Transition monoids
403:
18:Commutative monoid
6536:978-3-642-24896-2
6461:978-0-486-47189-1
6360:978-0-387-75450-5
6323:, pp. 3β28,
5830:Green's relations
5773:
5759:
5751:
5728:
5599:least upper bound
5583:continuous monoid
5468:
5428:
5393:
5391:
5361:
5322:
5305:
5271:
5201:
5150:
5112:
5010:In addition, any
4954:
4895:
4743:. Similarly, the
4683:
4682:
3832:for some strings
3106:absorbing element
2347:
2309:
2262:or, equivalently
1929:. The set of all
1866:identity function
1722:constant function
1672:Cartesian product
1600:Given any monoid
1558:as the operation.
1251:positive integers
1128:partially ordered
895:. In this case,
793:Monoid structures
670:, the equalities
487:syntactic monoids
383:
382:
16:(Redirected from
6744:
6737:Semigroup theory
6703:
6702:
6684:
6658:
6640:
6615:
6597:
6548:
6517:
6490:
6464:
6448:Jacobson, Nathan
6443:
6427:Jacobson, Nathan
6422:
6393:
6372:
6341:
6332:
6315:
6271:
6265:
6256:
6251:
6245:
6240:
6234:
6228:
6222:
6216:
6205:
6200:
6194:
6188:
6182:
6176:
6170:
6164:
6158:
6149:
6143:
6137:
6131:
6126:
6109:
6107:
6101:
6090:
6084:
6078:
6072:
6023:
6017:
6015:
6009:
5995:
5989:
5979:
5968:
5958:
5951:
5945:
5943:
5932:
5926:
5924:
5894:
5885:
5875:
5825:Cartesian monoid
5814:
5805:
5803:
5802:
5797:
5792:
5791:
5781:
5770:
5760:
5757:
5747:
5746:
5736:
5717:
5695:
5689:
5678:
5672:
5666:
5653:
5651:
5650:
5645:
5592:
5577:
5559:
5541:
5531:
5521:
5514:
5510:partial ordering
5508:together with a
5507:
5493:
5491:
5490:
5485:
5483:
5469:
5466:
5457:
5456:
5455:
5439:
5438:
5429:
5426:
5418:
5417:
5407:
5392:
5389:
5386:
5385:
5375:
5357:
5356:
5355:
5354:
5343:
5342:
5341:
5319:
5291:
5289:
5288:
5283:
5272:
5269:
5266:
5265:
5253:
5252:
5240:
5239:
5238:
5227:
5196:
5195:
5183:
5182:
5181:
5170:
5139:
5138:
5137:
5126:
5101:
5092:
5090:
5089:
5084:
5082:
5081:
5054:Complete monoids
5006:
5004:
5003:
4998:
4996:
4995:
4989:
4976:
4956:
4952:
4947:
4938:
4914:
4897:
4893:
4862:
4861:
4849:
4821:
4817:
4811:
4720:
4716:
4710:
4182:
4178:
4170:
4164:
4158:
4152:
4146:
4140:
4134:
4132:
4131:
4126:
4124:
4123:
4105:
4104:
4095:
4094:
4078:
4067:
4065:
4064:
4059:
3975:
3973:
3972:
3967:
3923:
3884:
3875:
3869:
3850:
3831:
3821:
3811:
3796:
3789:
3779:
3769:
3764:binary relations
3761:
3754:
3717:
3703:
3690:
3672:
3658:
3648:
3634:
3626:
3616:
3599:monoid morphisms
3596:
3590:
3584:
3573:
3558:
3533:
3527:
3521:
3515:
3481:
3467:
3459:
3443:
3442:
3433:
3432:
3423:
3386:
3372:
3346:
3340:
3334:
3328:
3322:
3313:
3299:
3293:
3283:
3268:
3262:
3254:
3248:. Then a (left)
3247:
3241:
3237:
3217:
3210:
3203:
3183:
3165:
3147:
3141:
3135:
3129:
3116:Types of monoids
3103:
3093:
3087:
3081:
3048:
3044:
3033:
3023:
3005:
2995:
2977:
2971:
2965:
2959:
2949:
2938:
2928:
2922:
2916:
2903:exist such that
2902:
2896:
2869:
2855:
2841:
2834:
2824:
2818:
2812:
2803:
2769:
2764:are inverses of
2763:
2757:
2751:
2745:
2739:
2725:
2711:
2701:
2687:
2676:
2662:
2655:
2641:
2634:
2625:
2614:
2584:
2571:
2565:
2546:of any sequence
2545:
2543:
2542:
2537:
2534:
2533:
2523:
2518:
2499:
2498:
2482:
2468:
2454:
2448:
2442:
2428:
2418:
2410:
2404:
2394:
2393:
2391:
2380:
2378:
2377:
2372:
2370:
2369:
2348:
2345:
2310:
2307:
2261:
2259:
2258:
2253:
2251:
2250:
2147:
2139:
2131:
2125:
2119:
2108:
2107:
2101:
2094:
2092:
2086:
2080:
2074:
2072:
2063:
2057:
2056:
2054:
2042:
2027:
2023:
2016:
2010:
1996:
1990:
1980:
1969:compact surfaces
1952:
1938:
1928:
1922:
1912:
1903:
1897:
1891:
1885:
1879:
1859:
1849:
1836:
1830:
1822:
1811:
1776:
1766:
1752:
1746:
1742:
1729:
1719:
1713:
1704:
1682:
1669:
1653:
1647:
1638:
1620:
1614:
1605:
1596:
1592:
1582:
1570:
1546:
1540:
1522:rational numbers
1512:
1489:
1481:
1477:
1468:
1454:
1444:
1427:
1417:
1395:
1389:
1383:
1356:
1352:
1335:Boolean algebras
1331:Heyting algebras
1298:
1292:
1283:
1277:
1271:
1262:
1258:
1244:numerical monoid
1241:
1234:
1228:
1223:
1222:= {0, 1, 2, ...}
1206:
1194:
1182:
1149:
1143:
1137:
1121:
1115:
1105:
1099:
1093:
1087:
1081:
1075:
1069:
1059:
1045:
1040:if there exists
1039:
1029:
996:is said to be a
995:
989:
983:
977:
971:
965:
956:
950:
932:
925:
915:
906:
900:
894:
880:
866:
852:
846:
841:. Symbolically,
840:
834:
828:
822:
813:
777:
763:
736:
721:
708:identity element
697:
683:
669:
663:
657:
651:
643:Identity element
639:
611:
605:
599:
593:
576:
572:
558:binary operation
556:equipped with a
555:
463:computer science
435:
426:identity element
422:binary operation
407:abstract algebra
375:
368:
361:
150:Commutative ring
79:Rack and quandle
44:
43:
21:
6752:
6751:
6747:
6746:
6745:
6743:
6742:
6741:
6732:Category theory
6717:
6716:
6669:
6666:
6661:
6613:
6579:
6537:
6527:Springer-Verlag
6514:
6504:Springer-Verlag
6480:
6462:
6441:
6412:
6361:
6351:Springer-Verlag
6330:10.1.1.304.6152
6304:
6290:Category Theory
6280:
6275:
6274:
6266:
6259:
6252:
6248:
6241:
6237:
6233:, pp. 7β10
6229:
6225:
6217:
6208:
6201:
6197:
6189:
6185:
6177:
6173:
6165:
6161:
6150:
6146:
6138:
6134:
6127:
6123:
6118:
6113:
6112:
6103:
6100:
6092:
6086:
6080:
6074:
6062:
6045:
6025:
6024:
6020:
6016:has an inverse.
6011:
5997:
5991:
5981:
5970:
5960:
5954:
5952:
5948:
5939:
5933:
5929:
5923:
5916:
5909:
5902:
5896:
5893:
5887:
5884:
5878:
5876:
5872:
5867:
5821:
5810:
5787:
5783:
5777:
5756:
5755:
5742:
5738:
5732:
5726:
5723:
5722:
5716:
5706:
5697:
5691:
5683:
5674:
5668:
5658:
5609:
5606:
5605:
5595:directed subset
5593:in which every
5586:
5561:
5543:
5533:
5523:
5516:
5512:
5503:
5476:
5467: for
5465:
5448:
5447:
5443:
5434:
5430:
5427: and
5425:
5413:
5409:
5397:
5388:
5381:
5377:
5365:
5350:
5346:
5345:
5337:
5333:
5326:
5321:
5309:
5303:
5300:
5299:
5270: for
5268:
5261:
5257:
5248:
5244:
5234:
5230:
5229:
5205:
5191:
5187:
5177:
5173:
5172:
5154:
5133:
5129:
5128:
5116:
5110:
5107:
5106:
5097:
5077:
5073:
5071:
5068:
5067:
5060:complete monoid
5056:
5050:of the monoid.
5028:
4991:
4990:
4982:
4963:
4950:
4948:
4940:
4925:
4916:
4915:
4904:
4891:
4889:
4879:
4878:
4857:
4853:
4836:
4834:
4831:
4830:
4819:
4813:
4807:
4799:by employing a
4781:
4718:
4712:
4704:
4177:
4166:
4160:
4154:
4148:
4142:
4136:
4119:
4115:
4100:
4096:
4090:
4086:
4084:
4081:
4080:
4076:
3991:
3988:
3987:
3981:bicyclic monoid
3932:
3929:
3928:
3921:
3912:
3903:
3896:
3886:
3880:
3871:
3852:
3833:
3823:
3813:
3812:if and only if
3807:
3798:
3791:
3781:
3774:
3767:
3759:
3752:
3741:
3735:
3705:
3699:
3689:
3685:
3681:
3674:
3671:
3667:
3663:
3660:
3657:
3653:
3650:
3647:
3643:
3636:
3633:
3628:
3622:
3619:residue classes
3615:
3610:
3592:
3586:
3583:
3575:
3572:
3564:
3557:
3548:
3536:
3529:
3523:
3517:
3486:
3469:
3461:
3453:
3436:
3435:
3426:
3425:
3418:
3411:
3390:operator monoid
3382:
3348:
3342:
3336:
3330:
3324:
3318:
3301:
3295:
3289:
3270:
3264:
3258:
3250:
3243:
3239:
3233:
3230:
3224:
3212:
3205:
3194:
3167:
3149:
3143:
3137:
3131:
3125:
3118:
3095:
3089:
3083:
3065:
3046:
3042:
3025:
3007:
2997:
2979:
2978:, the equality
2973:
2967:
2961:
2955:
2943:
2930:
2929:would get that
2924:
2918:
2904:
2898:
2892:
2889:
2883:
2857:
2843:
2836:
2826:
2820:
2814:
2808:
2771:
2765:
2759:
2753:
2747:
2741:
2727:
2713:
2707:
2697:
2694:
2678:
2664:
2657:
2643:
2636:
2630:
2616:
2613:
2604:
2594:
2586:
2579:
2573:
2567:
2563:
2554:
2547:
2529:
2525:
2519:
2508:
2494:
2490:
2488:
2485:
2484:
2478:
2475:
2456:
2450:
2444:
2438:
2435:
2424:
2413:{0, 1, 2, ...,
2412:
2406:
2399:
2387:
2385:
2384:
2365:
2364:
2344:
2342:
2333:
2332:
2306:
2304:
2285:
2284:
2267:
2264:
2263:
2245:
2244:
2239:
2228:
2223:
2218:
2213:
2207:
2206:
2195:
2184:
2179:
2174:
2169:
2159:
2158:
2156:
2153:
2152:
2142:{0, 1, 2, ...,
2141:
2135:
2133:
2127:
2121:
2110:
2103:
2097:
2096:
2088:
2082:
2076:
2068:
2066:
2065:
2059:
2050:
2048:
2047:
2029:
2025:
2018:
2012:
1998:
1992:
1986:
1976:
1946:
1940:
1934:
1924:
1918:
1908:
1899:
1898:is finite with
1893:
1887:
1886:is finite with
1881:
1875:
1851:
1845:
1832:
1824:
1813:
1778:
1772:
1757:
1748:
1744:
1738:
1725:
1715:
1709:
1703:
1694:
1688:
1674:
1667:
1661:
1655:
1649:
1643:
1642:Given two sets
1622:
1616:
1610:
1608:opposite monoid
1601:
1594:
1590:
1580:
1568:
1552:matrix addition
1542:
1536:
1535:The set of all
1530:complex numbers
1510:
1483:
1479:
1473:
1464:
1450:
1440:
1419:
1397:
1391:
1385:
1379:
1354:
1346:
1294:
1288:
1279:
1273:
1267:
1260:
1253:
1237:
1232:
1226:
1218:
1216:natural numbers
1204:
1192:
1180:
1172:
1156:
1145:
1139:
1133:
1117:
1107:
1101:
1095:
1094:, there exists
1089:
1083:
1077:
1071:
1065:
1047:
1041:
1031:
1027:
1006:
991:
985:
979:
973:
967:
961:
952:
946:
943:
930:
921:
911:
902:
896:
882:
868:
854:
848:
842:
836:
830:
824:
818:
807:
800:
795:
765:
745:
726:
719:
685:
671:
665:
659:
653:
647:
613:
612:, the equation
607:
601:
595:
589:
574:
560:
551:
545:
518:automata theory
503:process calculi
499:history monoids
452:category theory
433:
379:
350:
349:
348:
319:Non-associative
301:
290:
289:
279:
259:
248:
247:
236:Map of lattices
232:
228:Boolean algebra
223:Heyting algebra
197:
186:
185:
179:
160:Integral domain
124:
113:
112:
106:
60:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6750:
6740:
6739:
6734:
6729:
6715:
6714:
6704:
6685:
6665:
6664:External links
6662:
6660:
6659:
6631:(1): 267β285.
6616:
6611:
6598:
6577:
6555:, ed. (1997),
6549:
6535:
6518:
6512:
6491:
6478:
6465:
6460:
6444:
6439:
6423:
6410:
6398:Howie, John M.
6394:
6373:
6359:
6342:
6316:
6302:
6281:
6279:
6276:
6273:
6272:
6257:
6246:
6235:
6223:
6206:
6195:
6183:
6171:
6159:
6144:
6132:
6120:
6119:
6117:
6114:
6111:
6110:
6096:
6058:
6041:
6018:
5946:
5927:
5921:
5914:
5907:
5900:
5891:
5882:
5869:
5868:
5866:
5863:
5862:
5861:
5856:
5851:
5846:
5844:Kleene algebra
5837:
5832:
5827:
5820:
5817:
5807:
5806:
5795:
5790:
5786:
5780:
5776:
5769:
5766:
5763:
5754:
5750:
5745:
5741:
5735:
5731:
5708:
5702:
5655:
5654:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5496:
5495:
5482:
5479:
5475:
5472:
5463:
5460:
5454:
5451:
5446:
5442:
5437:
5433:
5424:
5421:
5416:
5412:
5406:
5403:
5400:
5396:
5390: if
5384:
5380:
5374:
5371:
5368:
5364:
5360:
5353:
5349:
5340:
5336:
5332:
5329:
5325:
5318:
5315:
5312:
5308:
5293:
5292:
5281:
5278:
5275:
5264:
5260:
5256:
5251:
5247:
5243:
5237:
5233:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5204:
5199:
5194:
5190:
5186:
5180:
5176:
5169:
5166:
5163:
5160:
5157:
5153:
5148:
5145:
5142:
5136:
5132:
5125:
5122:
5119:
5115:
5080:
5076:
5066:sum operation
5055:
5052:
5027:
5024:
5020:tree traversal
5012:data structure
5008:
5007:
4994:
4988:
4985:
4980:
4975:
4972:
4969:
4966:
4962:
4959:
4949:
4946:
4943:
4937:
4934:
4931:
4928:
4924:
4921:
4918:
4917:
4913:
4910:
4907:
4903:
4900:
4890:
4888:
4885:
4884:
4882:
4877:
4874:
4871:
4868:
4865:
4860:
4856:
4852:
4848:
4845:
4842:
4839:
4780:
4777:
4701:
4700:
4681:
4680:
4677:
4674:
4671:
4668:
4665:
4659:
4658:
4655:
4652:
4649:
4646:
4643:
4637:
4636:
4633:
4630:
4627:
4624:
4621:
4615:
4614:
4611:
4608:
4605:
4602:
4599:
4593:
4592:
4589:
4586:
4583:
4580:
4577:
4570:
4569:
4566:
4563:
4560:
4557:
4554:
4547:
4546:
4543:
4540:
4537:
4534:
4531:
4524:
4523:
4520:
4517:
4514:
4511:
4508:
4502:
4501:
4498:
4495:
4492:
4489:
4486:
4479:
4478:
4475:
4472:
4469:
4466:
4463:
4457:
4456:
4453:
4450:
4447:
4444:
4441:
4434:
4433:
4430:
4427:
4424:
4421:
4418:
4412:
4411:
4408:
4405:
4402:
4399:
4396:
4389:
4388:
4385:
4382:
4379:
4376:
4373:
4367:
4366:
4363:
4360:
4357:
4354:
4351:
4344:
4343:
4340:
4337:
4334:
4331:
4328:
4322:
4321:
4318:
4315:
4312:
4309:
4306:
4299:
4298:
4295:
4292:
4289:
4286:
4283:
4277:
4276:
4273:
4270:
4267:
4264:
4261:
4259:Small category
4255:
4254:
4251:
4248:
4245:
4242:
4239:
4233:
4232:
4229:
4226:
4223:
4220:
4217:
4211:
4210:
4205:
4200:
4195:
4190:
4185:
4176:
4173:
4122:
4118:
4114:
4111:
4108:
4103:
4099:
4093:
4089:
4073:plactic monoid
4069:
4068:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4008:
4004:
4001:
3998:
3995:
3977:
3976:
3965:
3962:
3959:
3956:
3953:
3949:
3945:
3942:
3939:
3936:
3917:
3908:
3901:
3894:
3803:
3737:Main article:
3734:
3731:
3687:
3683:
3679:
3669:
3665:
3661:
3655:
3651:
3645:
3641:
3629:
3611:
3579:
3568:
3561:
3560:
3553:
3544:
3534:
3468:is a function
3410:
3407:
3375:
3374:
3315:
3226:Main article:
3223:
3220:
3122:inverse monoid
3117:
3114:
2885:Main article:
2882:
2879:
2740:. The element
2693:
2690:
2609:
2599:
2590:
2577:
2559:
2552:
2532:
2528:
2522:
2517:
2514:
2511:
2507:
2502:
2497:
2493:
2474:
2471:
2443:is unique: If
2434:
2431:
2368:
2363:
2360:
2357:
2354:
2351:
2343:
2341:
2338:
2335:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2305:
2303:
2300:
2297:
2294:
2291:
2290:
2288:
2283:
2280:
2277:
2274:
2271:
2249:
2243:
2240:
2238:
2235:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2208:
2205:
2202:
2199:
2196:
2194:
2191:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2164:
2162:
2150:
2149:
2044:
1958:
1942:
1905:
1842:
1735:
1706:
1699:
1692:
1685:direct product
1665:
1659:
1640:
1598:
1561:
1560:
1559:
1533:
1503:
1502:
1501:
1500:
1499:
1373:
1362:
1340:
1339:
1338:
1304:
1285:
1264:
1247:
1212:
1171:
1168:
1155:
1152:
1070:is an element
1018:abelian monoid
1005:
1002:
942:
939:
799:
796:
794:
791:
700:
699:
644:
641:
586:
544:
541:
409:, a branch of
401:with identity.
381:
380:
378:
377:
370:
363:
355:
352:
351:
347:
346:
341:
336:
331:
326:
321:
316:
310:
309:
308:
302:
296:
295:
292:
291:
288:
287:
284:Linear algebra
278:
277:
272:
267:
261:
260:
254:
253:
250:
249:
246:
245:
242:Lattice theory
238:
231:
230:
225:
220:
215:
210:
205:
199:
198:
192:
191:
188:
187:
178:
177:
172:
167:
162:
157:
152:
147:
142:
137:
132:
126:
125:
119:
118:
115:
114:
105:
104:
99:
94:
88:
87:
86:
81:
76:
67:
61:
55:
54:
51:
50:
26:
9:
6:
4:
3:
2:
6749:
6738:
6735:
6733:
6730:
6728:
6725:
6724:
6722:
6712:
6708:
6705:
6700:
6699:
6694:
6691:
6686:
6682:
6678:
6677:
6672:
6668:
6667:
6656:
6652:
6648:
6644:
6639:
6634:
6630:
6626:
6622:
6617:
6614:
6612:9780387097817
6608:
6604:
6599:
6596:
6592:
6588:
6584:
6580:
6578:0-521-59924-5
6574:
6570:
6566:
6562:
6558:
6554:
6550:
6546:
6542:
6538:
6532:
6528:
6524:
6519:
6515:
6513:3-540-52826-1
6509:
6505:
6501:
6497:
6492:
6489:
6485:
6481:
6479:3-11-015248-7
6475:
6471:
6466:
6463:
6457:
6453:
6452:Basic algebra
6449:
6445:
6442:
6440:0-387-90122-1
6436:
6432:
6428:
6424:
6421:
6417:
6413:
6411:0-19-851194-9
6407:
6403:
6399:
6395:
6391:
6387:
6383:
6380:(in German).
6379:
6374:
6370:
6366:
6362:
6356:
6352:
6348:
6343:
6340:
6336:
6331:
6326:
6322:
6317:
6313:
6309:
6305:
6303:0-19-856861-4
6299:
6295:
6291:
6287:
6286:Awodey, Steve
6283:
6282:
6269:
6264:
6262:
6255:
6250:
6244:
6239:
6232:
6227:
6220:
6215:
6213:
6211:
6204:
6199:
6192:
6191:Jacobson 2009
6187:
6180:
6179:Jacobson 2009
6175:
6168:
6167:Jacobson 2009
6163:
6157:
6153:
6148:
6141:
6136:
6130:
6129:Jacobson 2009
6125:
6121:
6106:
6099:
6095:
6089:
6083:
6077:
6070:
6066:
6061:
6057:
6053:
6049:
6044:
6040:
6036:
6032:
6028:
6022:
6014:
6008:
6004:
6000:
5994:
5988:
5984:
5977:
5973:
5967:
5963:
5957:
5950:
5942:
5937:
5931:
5920:
5913:
5906:
5899:
5890:
5881:
5874:
5870:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5841:
5838:
5836:
5833:
5831:
5828:
5826:
5823:
5822:
5816:
5813:
5793:
5788:
5784:
5778:
5774:
5767:
5764:
5761:
5748:
5743:
5739:
5733:
5729:
5721:
5720:
5719:
5715:
5711:
5705:
5701:
5694:
5687:
5680:
5677:
5671:
5665:
5661:
5638:
5635:
5632:
5623:
5620:
5614:
5611:
5604:
5603:
5602:
5600:
5596:
5590:
5584:
5579:
5576:
5572:
5568:
5564:
5558:
5554:
5550:
5546:
5540:
5536:
5530:
5526:
5519:
5511:
5506:
5501:
5480:
5477:
5473:
5470:
5458:
5452:
5449:
5444:
5440:
5435:
5431:
5422:
5419:
5414:
5410:
5404:
5401:
5398:
5394:
5382:
5378:
5372:
5369:
5366:
5362:
5358:
5351:
5347:
5338:
5334:
5330:
5327:
5323:
5316:
5313:
5310:
5306:
5298:
5297:
5296:
5279:
5276:
5273:
5262:
5258:
5254:
5249:
5245:
5241:
5235:
5231:
5221:
5218:
5215:
5209:
5206:
5202:
5197:
5192:
5188:
5184:
5178:
5174:
5164:
5158:
5155:
5151:
5146:
5143:
5140:
5134:
5130:
5120:
5117:
5113:
5105:
5104:
5103:
5100:
5096:
5078:
5065:
5061:
5051:
5048:
5044:
5039:
5037:
5033:
5023:
5021:
5017:
5013:
4986:
4983:
4978:
4960:
4957:
4944:
4941:
4922:
4919:
4901:
4898:
4886:
4880:
4872:
4869:
4866:
4858:
4854:
4850:
4829:
4828:
4827:
4825:
4816:
4810:
4804:
4802:
4798:
4794:
4790:
4786:
4776:
4774:
4773:
4768:
4767:monoid object
4763:
4761:
4756:
4752:
4750:
4746:
4742:
4738:
4735:
4731:
4727:
4722:
4715:
4708:
4699:
4696:
4695:
4694:
4692:
4688:
4678:
4675:
4672:
4669:
4666:
4664:
4663:Abelian group
4661:
4660:
4656:
4653:
4650:
4647:
4644:
4642:
4639:
4638:
4634:
4631:
4628:
4625:
4622:
4620:
4617:
4616:
4612:
4609:
4606:
4603:
4600:
4598:
4595:
4594:
4590:
4587:
4584:
4581:
4578:
4576:
4572:
4571:
4567:
4564:
4561:
4558:
4555:
4553:
4549:
4548:
4544:
4541:
4538:
4535:
4532:
4530:
4526:
4525:
4521:
4518:
4515:
4512:
4509:
4507:
4504:
4503:
4499:
4496:
4493:
4490:
4487:
4485:
4481:
4480:
4476:
4473:
4470:
4467:
4464:
4462:
4459:
4458:
4454:
4451:
4448:
4445:
4442:
4440:
4436:
4435:
4431:
4428:
4425:
4422:
4419:
4417:
4414:
4413:
4409:
4406:
4403:
4400:
4397:
4395:
4391:
4390:
4386:
4383:
4380:
4377:
4374:
4372:
4369:
4368:
4364:
4361:
4358:
4355:
4352:
4350:
4346:
4345:
4341:
4338:
4335:
4332:
4329:
4327:
4324:
4323:
4319:
4316:
4313:
4310:
4307:
4305:
4301:
4300:
4296:
4293:
4290:
4287:
4284:
4282:
4279:
4278:
4274:
4271:
4268:
4265:
4262:
4260:
4257:
4256:
4252:
4249:
4246:
4243:
4240:
4238:
4235:
4234:
4230:
4227:
4224:
4221:
4218:
4216:
4215:Partial magma
4213:
4212:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4183:
4172:
4169:
4163:
4157:
4151:
4145:
4139:
4135:for integers
4120:
4112:
4109:
4101:
4097:
4091:
4087:
4074:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4002:
3999:
3996:
3986:
3985:
3984:
3982:
3960:
3957:
3954:
3951:
3943:
3940:
3937:
3927:
3926:
3925:
3920:
3916:
3911:
3907:
3900:
3893:
3889:
3883:
3877:
3874:
3868:
3864:
3860:
3856:
3848:
3844:
3840:
3836:
3830:
3826:
3820:
3816:
3810:
3806:
3801:
3794:
3788:
3784:
3777:
3771:
3765:
3758:
3750:
3746:
3740:
3730:
3728:
3724:
3719:
3716:
3712:
3708:
3702:
3697:
3692:
3677:
3639:
3632:
3625:
3620:
3617:, the set of
3614:
3607:
3602:
3600:
3595:
3589:
3582:
3578:
3571:
3567:
3556:
3552:
3547:
3543:
3539:
3535:
3532:
3526:
3520:
3513:
3509:
3505:
3501:
3497:
3493:
3489:
3485:
3484:
3483:
3480:
3476:
3472:
3465:
3457:
3451:
3440:
3430:
3421:
3415:
3406:
3404:
3400:
3396:
3392:
3391:
3385:
3380:
3371:
3367:
3363:
3359:
3355:
3351:
3345:
3339:
3333:
3327:
3321:
3316:
3312:
3308:
3304:
3298:
3292:
3287:
3286:
3285:
3282:
3278:
3274:
3267:
3261:
3256:
3253:
3246:
3236:
3229:
3219:
3215:
3208:
3204:implies that
3201:
3197:
3192:
3191:
3185:
3182:
3178:
3174:
3170:
3164:
3160:
3156:
3152:
3146:
3140:
3134:
3128:
3123:
3113:
3111:
3110:trivial group
3107:
3102:
3098:
3092:
3086:
3080:
3076:
3072:
3068:
3061:
3057:
3055:
3050:
3040:
3035:
3032:
3028:
3022:
3018:
3014:
3010:
3004:
3000:
2994:
2990:
2986:
2982:
2976:
2970:
2964:
2958:
2953:
2947:
2940:
2937:
2933:
2927:
2921:
2915:
2911:
2907:
2901:
2895:
2888:
2878:
2876:
2871:
2868:
2864:
2860:
2856:hold for all
2854:
2850:
2846:
2839:
2833:
2829:
2823:
2817:
2811:
2805:
2802:
2798:
2794:
2790:
2786:
2782:
2778:
2774:
2768:
2762:
2756:
2750:
2744:
2738:
2734:
2730:
2724:
2720:
2716:
2710:
2705:
2700:
2689:
2685:
2681:
2675:
2671:
2667:
2660:
2654:
2650:
2646:
2639:
2635:of a monoid:
2633:
2627:
2624:
2620:
2612:
2608:
2602:
2598:
2593:
2589:
2583:
2576:
2570:
2562:
2558:
2551:
2530:
2526:
2520:
2515:
2512:
2509:
2505:
2500:
2495:
2491:
2481:
2470:
2467:
2463:
2459:
2453:
2447:
2441:
2430:
2427:
2422:
2416:
2409:
2402:
2396:
2390:
2381:
2361:
2358:
2355:
2352:
2349:
2339:
2336:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2301:
2298:
2295:
2292:
2286:
2281:
2275:
2269:
2247:
2241:
2236:
2233:
2230:
2225:
2220:
2215:
2210:
2203:
2200:
2197:
2192:
2189:
2186:
2181:
2176:
2171:
2166:
2160:
2145:
2138:
2130:
2124:
2118:
2114:
2106:
2100:
2091:
2085:
2079:
2071:
2062:
2053:
2045:
2041:
2037:
2033:
2021:
2015:
2009:
2005:
2001:
1995:
1989:
1984:
1979:
1974:
1973:connected sum
1970:
1966:
1963:
1962:homeomorphism
1959:
1956:
1950:
1945:
1937:
1932:
1931:endomorphisms
1927:
1923:an object of
1921:
1916:
1911:
1906:
1902:
1896:
1890:
1884:
1878:
1873:
1872:
1867:
1863:
1858:
1854:
1848:
1843:
1840:
1835:
1828:
1820:
1816:
1812:. This turns
1809:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1775:
1770:
1764:
1760:
1756:
1751:
1741:
1737:Fix a monoid
1736:
1733:
1728:
1723:
1718:
1712:
1708:Fix a monoid
1707:
1702:
1698:
1691:
1686:
1681:
1677:
1673:
1668:
1658:
1652:
1646:
1641:
1637:
1633:
1629:
1625:
1619:
1613:
1609:
1604:
1599:
1588:
1587:
1578:
1574:
1566:
1562:
1557:
1553:
1549:
1545:
1539:
1534:
1531:
1527:
1523:
1519:
1515:
1514:
1508:
1504:
1497:
1493:
1487:
1476:
1471:
1470:
1467:
1462:
1458:
1453:
1448:
1443:
1438:
1434:
1433:
1431:
1426:
1422:
1416:
1412:
1408:
1404:
1400:
1396:and defining
1394:
1388:
1382:
1378:
1374:
1371:
1370:abelian group
1367:
1363:
1360:
1359:trivial group
1350:
1345:
1344:singleton set
1341:
1336:
1332:
1328:
1324:
1320:
1316:
1315:
1313:
1309:
1305:
1302:
1297:
1291:
1286:
1282:
1276:
1270:
1265:
1256:
1252:
1248:
1245:
1240:
1235:
1229:
1221:
1217:
1213:
1210:
1202:
1198:
1190:
1186:
1181:{False, True}
1178:
1174:
1173:
1167:
1165:
1161:
1151:
1148:
1142:
1136:
1132:
1131:abelian group
1129:
1125:
1124:positive cone
1120:
1114:
1110:
1104:
1098:
1092:
1086:
1080:
1074:
1068:
1063:
1058:
1054:
1050:
1044:
1038:
1034:
1030:, defined by
1026:
1023:
1019:
1015:
1011:
1001:
999:
994:
988:
982:
976:
970:
964:
960:
955:
949:
938:
936:
929:
928:singleton set
924:
919:
914:
908:
905:
899:
893:
889:
885:
879:
875:
871:
865:
861:
857:
851:
845:
839:
833:
827:
821:
817:
811:
805:
790:
788:
784:
779:
776:
772:
768:
761:
757:
753:
749:
743:
742:juxtaposition
738:
734:
730:
725:
717:
713:
709:
705:
696:
692:
688:
682:
678:
674:
668:
662:
656:
650:
645:
642:
637:
633:
629:
625:
621:
617:
610:
604:
598:
592:
587:
585:Associativity
584:
583:
582:
580:
571:
567:
563:
559:
554:
550:
540:
538:
533:
531:
527:
523:
519:
515:
510:
508:
504:
500:
496:
495:Trace monoids
492:
488:
484:
480:
476:
472:
469:, the set of
468:
464:
459:
457:
453:
448:
446:
442:
437:
431:
427:
423:
420:
416:
412:
408:
400:
396:
392:
387:
376:
371:
369:
364:
362:
357:
356:
354:
353:
345:
342:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
311:
307:
304:
303:
299:
294:
293:
286:
285:
281:
280:
276:
273:
271:
268:
266:
263:
262:
257:
252:
251:
244:
243:
239:
237:
234:
233:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
200:
195:
190:
189:
184:
183:
176:
173:
171:
170:Division ring
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
127:
122:
117:
116:
111:
110:
103:
100:
98:
95:
93:
92:Abelian group
90:
89:
85:
82:
80:
77:
75:
71:
68:
66:
63:
62:
58:
53:
52:
49:
46:
45:
40:
33:
19:
6696:
6674:
6628:
6624:
6602:
6556:
6553:Lothaire, M.
6522:
6499:
6469:
6451:
6430:
6401:
6381:
6377:
6346:
6320:
6289:
6249:
6243:Hebisch 1992
6238:
6226:
6221:, p. 10
6203:Wehrung 1996
6198:
6186:
6181:, p. 35
6174:
6162:
6147:
6142:, p. 13
6135:
6124:
6104:
6097:
6093:
6087:
6081:
6075:
6068:
6064:
6059:
6055:
6051:
6047:
6042:
6038:
6034:
6030:
6026:
6021:
6012:
6006:
6002:
5998:
5992:
5986:
5982:
5975:
5971:
5965:
5961:
5955:
5949:
5940:
5935:
5930:
5918:
5911:
5904:
5897:
5888:
5879:
5873:
5854:Vedic square
5811:
5808:
5758:finite
5713:
5709:
5703:
5699:
5692:
5685:
5681:
5675:
5669:
5663:
5659:
5656:
5588:
5582:
5580:
5574:
5570:
5566:
5562:
5556:
5552:
5548:
5544:
5538:
5534:
5528:
5524:
5517:
5504:
5499:
5497:
5294:
5098:
5059:
5057:
5040:
5029:
5009:
4823:
4814:
4808:
4805:
4797:parallelized
4782:
4771:
4764:
4759:
4757:
4753:
4748:
4740:
4736:
4732:between the
4723:
4713:
4706:
4702:
4697:
4684:
4596:
4550:Associative
4527:Commutative
4482:Commutative
4439:unital magma
4437:Commutative
4416:Unital magma
4392:Commutative
4347:Commutative
4302:Commutative
4237:Semigroupoid
4203:Cancellation
4167:
4161:
4155:
4149:
4143:
4137:
4070:
3978:
3918:
3914:
3909:
3905:
3898:
3891:
3887:
3881:
3878:
3872:
3866:
3862:
3858:
3854:
3846:
3842:
3838:
3834:
3828:
3824:
3818:
3814:
3808:
3804:
3799:
3797:by defining
3792:
3786:
3782:
3775:
3772:
3745:presentation
3744:
3742:
3720:
3714:
3710:
3706:
3700:
3693:
3675:
3637:
3630:
3623:
3612:
3603:
3598:
3593:
3587:
3580:
3576:
3569:
3565:
3562:
3554:
3550:
3545:
3541:
3537:
3530:
3524:
3518:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3478:
3474:
3470:
3463:
3455:
3450:homomorphism
3447:
3438:
3428:
3419:
3399:semiautomata
3388:
3383:
3379:group action
3376:
3369:
3365:
3361:
3357:
3353:
3349:
3343:
3337:
3331:
3325:
3319:
3310:
3306:
3302:
3296:
3290:
3280:
3276:
3272:
3265:
3259:
3251:
3249:
3244:
3234:
3231:
3213:
3206:
3199:
3195:
3188:
3186:
3180:
3176:
3172:
3168:
3162:
3158:
3154:
3150:
3144:
3138:
3132:
3126:
3121:
3119:
3100:
3096:
3090:
3084:
3078:
3074:
3070:
3066:
3062:
3058:
3053:
3051:
3038:
3036:
3030:
3026:
3020:
3016:
3012:
3008:
3002:
2998:
2992:
2988:
2984:
2980:
2974:
2968:
2962:
2956:
2945:
2941:
2935:
2931:
2925:
2919:
2913:
2909:
2905:
2899:
2893:
2890:
2872:
2866:
2862:
2858:
2852:
2848:
2844:
2837:
2831:
2827:
2821:
2815:
2809:
2806:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2766:
2760:
2754:
2748:
2742:
2736:
2732:
2728:
2722:
2718:
2714:
2708:
2698:
2695:
2683:
2679:
2673:
2669:
2665:
2658:
2652:
2648:
2644:
2637:
2631:
2628:
2622:
2618:
2610:
2606:
2600:
2596:
2591:
2587:
2581:
2574:
2568:
2560:
2556:
2549:
2479:
2476:
2465:
2461:
2457:
2451:
2445:
2439:
2436:
2425:
2421:cyclic group
2414:
2407:
2400:
2397:
2388:
2382:
2151:
2143:
2136:
2128:
2122:
2120:. Each such
2116:
2112:
2104:
2098:
2089:
2083:
2077:
2069:
2060:
2051:
2039:
2035:
2031:
2019:
2013:
2007:
2003:
1999:
1993:
1987:
1977:
1948:
1943:
1935:
1925:
1919:
1909:
1900:
1894:
1888:
1882:
1876:
1869:
1856:
1852:
1846:
1833:
1826:
1818:
1814:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1773:
1762:
1758:
1749:
1739:
1726:
1716:
1710:
1700:
1696:
1689:
1679:
1675:
1663:
1656:
1650:
1644:
1635:
1631:
1627:
1623:
1617:
1611:
1607:
1602:
1584:
1577:empty string
1543:
1537:
1526:real numbers
1495:
1485:
1474:
1465:
1456:
1451:
1441:
1436:
1430:free functor
1424:
1420:
1414:
1410:
1406:
1402:
1398:
1392:
1386:
1380:
1348:
1295:
1289:
1287:Given a set
1280:
1274:
1268:
1266:Given a set
1254:
1238:
1219:
1160:trace monoid
1157:
1146:
1140:
1134:
1118:
1112:
1108:
1102:
1096:
1090:
1084:
1078:
1072:
1066:
1061:
1056:
1052:
1048:
1042:
1036:
1032:
1021:
1017:
1013:
1012:is called a
1007:
997:
992:
986:
980:
974:
968:
962:
958:
953:
947:
944:
922:
912:
909:
903:
897:
891:
887:
883:
877:
873:
869:
863:
859:
855:
849:
843:
837:
831:
825:
819:
809:
806:of a monoid
803:
801:
780:
774:
770:
766:
759:
755:
751:
747:
739:
732:
728:
701:
694:
690:
686:
680:
676:
672:
666:
660:
654:
648:
635:
631:
627:
623:
619:
615:
608:
602:
596:
590:
578:
569:
565:
561:
552:
546:
534:
511:
460:
449:
439:Monoids are
438:
414:
404:
344:Hopf algebra
282:
275:Vector space
240:
180:
109:Group theory
107:
73:
72: /
6219:Awodey 2006
5016:binary tree
4730:equivalence
4208:Commutative
4193:Associative
3757:free monoid
3727:isomorphism
3673:. However,
3263:) is a set
2825:by setting
2696:An element
2064:, that is,
1960:The set of
1586:free monoid
1490:models the
1449:over a set
1308:semilattice
1257:∖ {0}
1249:The set of
1214:The set of
1025:preordering
1010:commutative
972:containing
957:is said to
479:free monoid
419:associative
411:mathematics
329:Lie algebra
314:Associative
218:Total order
208:Semilattice
182:Ring theory
6721:Categories
6711:PlanetMath
6655:0865.06010
6595:0874.20040
6545:1251.68135
6488:0945.20036
6420:0835.20077
6390:0747.08005
6384:: 21β152.
6369:1201.16038
6312:1100.18001
6278:References
6268:Kuich 2011
6254:Kuich 1990
6154:, p.
5859:Frobenioid
5657:for every
5522:for every
5515:such that
5102:such that
5064:infinitary
4801:prefix sum
4687:categories
4575:quasigroup
4552:quasigroup
4394:quasigroup
4371:Quasigroup
4075:of degree
3704:such that
3604:Not every
3482:such that
3228:monoid act
3148:such that
2712:such that
2704:invertible
2702:is called
2433:Properties
2134:Moreover,
2028:. We have
1939:, denoted
1437:find-first
1312:idempotent
1209:idempotent
1106:such that
1062:order-unit
1046:such that
941:Generators
798:Submonoids
543:Definition
475:characters
441:semigroups
399:semigroups
6698:MathWorld
6681:EMS Press
6325:CiteSeerX
6116:Citations
6073:for each
5969:for some
5775:∑
5765:⊂
5730:∑
5474:≠
5462:∅
5441:∩
5402:∈
5395:⋃
5370:∈
5363:∑
5331:∈
5324:∑
5314:∈
5307:∑
5277:≠
5210:∈
5203:∑
5159:∈
5152:∑
5124:∅
5121:∈
5114:∑
5095:index set
5075:Σ
5032:MapReduce
5026:MapReduce
4984:ℓ
4958:ℓ
4942:ℓ
4923:∙
4899:ℓ
4887:ε
4876:↦
4873:ℓ
4864:→
4859:∗
4679:Required
4657:Unneeded
4635:Required
4613:Unneeded
4591:Required
4568:Unneeded
4545:Required
4529:semigroup
4522:Unneeded
4506:Semigroup
4500:Required
4477:Unneeded
4455:Required
4432:Unneeded
4410:Required
4387:Unneeded
4365:Required
4342:Unneeded
4320:Required
4297:Unneeded
4275:Unneeded
4253:Unneeded
4231:Unneeded
4056:⟩
3994:⟨
3964:⟩
3935:⟨
3723:bijective
3649:given by
3271:β
:
2942:A monoid
2835:for each
2506:∏
2423:of order
2359:−
2327:−
2315:≤
2234:−
2226:⋯
2201:−
2190:−
2182:⋯
2109:for some
1971:with the
1955:morphisms
1904:elements.
1755:power set
1732:pointwise
1670:), their
1457:find-last
1377:semigroup
1301:empty set
1022:algebraic
945:A subset
881:whenever
804:submonoid
704:semigroup
537:semigroup
339:Bialgebra
145:Near-ring
102:Lie group
70:Semigroup
6693:"Monoid"
6671:"Monoid"
6647:56410568
6450:(2009),
6429:(1951),
6400:(1995),
6288:(2006).
5877:If both
5840:Semiring
5819:See also
5560:for all
5542:implies
5481:′
5453:′
5093:for any
5047:multiset
5043:multiset
4987:′
4953:if
4945:′
4894:if
4789:sequence
4726:functors
4691:morphism
4676:Required
4673:Required
4670:Required
4667:Required
4654:Required
4651:Required
4648:Required
4645:Required
4632:Unneeded
4629:Required
4626:Required
4623:Required
4610:Unneeded
4607:Required
4604:Required
4601:Required
4588:Required
4585:Unneeded
4582:Required
4579:Required
4565:Required
4562:Unneeded
4559:Required
4556:Required
4542:Unneeded
4539:Unneeded
4536:Required
4533:Required
4519:Unneeded
4516:Unneeded
4513:Required
4510:Required
4497:Required
4494:Required
4491:Unneeded
4488:Required
4474:Required
4471:Required
4468:Unneeded
4465:Required
4452:Unneeded
4449:Required
4446:Unneeded
4443:Required
4429:Unneeded
4426:Required
4423:Unneeded
4420:Required
4407:Required
4404:Unneeded
4401:Unneeded
4398:Required
4384:Required
4381:Unneeded
4378:Unneeded
4375:Required
4362:Unneeded
4359:Unneeded
4356:Unneeded
4353:Required
4339:Unneeded
4336:Unneeded
4333:Unneeded
4330:Required
4317:Required
4314:Required
4311:Required
4308:Unneeded
4304:Groupoid
4294:Required
4291:Required
4288:Required
4285:Unneeded
4281:Groupoid
4272:Unneeded
4269:Required
4266:Required
4263:Unneeded
4250:Unneeded
4247:Unneeded
4244:Required
4241:Unneeded
4228:Unneeded
4225:Unneeded
4222:Unneeded
4219:Unneeded
4198:Identity
3640: :
3516:for all
3473: :
3381:. Right
3317:for all
3288:for all
3024:implies
2996:implies
2950:has the
2677:for all
2663:. Then
2585:and let
2392:⟩
2386:⟨
2346:if
2308:if
2148:given by
2073:⟩
2067:⟨
2055:⟩
2049:⟨
1915:category
1794: :
1695:Γ β
β
β
Γ
1548:matrices
1518:integers
1480:{lt, gt}
1418:for all
1325:- and a
1284:itself).
1170:Examples
959:generate
718:, i. e.
716:constant
706:with an
588:For all
430:integers
175:Lie ring
140:Semiring
6683:, 2001
6587:1475463
6085:, when
4188:Closure
4071:is the
3904:, ...,
3795:β Ξ£ Γ Ξ£
3778:β Ξ£ Γ Ξ£
3621:modulo
3441:, Γ, 1)
3431:, +, 0)
3082:, then
2555:, ...,
2095:. Then
2087:, ...,
1965:classes
1769:subsets
1662:, ...,
1565:strings
1445:to the
1390:not in
1319:lattice
1122:is the
990:, then
783:inverse
577:, is a
524:), and
471:strings
424:and an
306:Algebra
298:Algebra
203:Lattice
194:Lattice
6707:Monoid
6653:
6645:
6609:
6593:
6585:
6575:
6543:
6533:
6510:
6486:
6476:
6458:
6437:
6418:
6408:
6388:
6367:
6357:
6327:
6310:
6300:
5990:where
5978:> 0
5597:has a
5532:, and
4822:, the
4793:folded
4597:Monoid
3983:, and
3563:where
3054:finite
2022:= 0, 1
2011:where
1985:, and
1606:, the
1364:Every
1342:Every
1310:is an
1195:while
918:closed
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