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# Topic: Monoid

 Monoid - Wikipedia, the free encyclopedia In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). The axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms which start and end at a given object (i.e. en.wikipedia.org /wiki/Monoid   (1059 words)

 [No title] This example represents the monoid which has as elements all positive and negative even numbers, the monoid operation is binary addition, inverses are the negative of the element and the identity is the zero element. Its argument should be a monoid M. When applied to M, it denotes the submonoid of M consisting of all invertible elements in M. This is a group. Its first argument should be a monoid M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a divisor of b in M. This means that there are u,v in carrier(M) such that uav=b. www.win.tue.nl /~amc/oz/om/cds/monoid1.html   (472 words)

 Monoid ring - Wikipedia, the free encyclopedia In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. Let R be a ring and G be a monoid. The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G]. en.wikipedia.org /wiki/Monoid_ring   (281 words)

 Monoid - FreeEncyclopedia   (Site not responding. Last check: 2007-11-06) A monoid is a pair (M,*), where M is a set and * is a binary operation on M, obeying the following rules: In other words, a monoid is a semigroup with an identity element. It is possible to view categories as generalizations of monoids: the composition of morphism in a category shares all properties of a monoid operation except that not all pairs of morphisms may be composed. openproxy.ath.cx /mo/Monoid.html   (626 words)

 GAP Manual: 87 Monoids and Semigroups Then the functions which construct monoids and semigroups and the functions which test whether a given object is a monoid or a semigroup are described (see SemiGroup, IsSemiGroup, Monoid and IsMonoid). Monoids and semigroups are domains, so every set theoretic function for domains is applicable to them (see Set Functions for Monoids). Standard monoid elements may be compared with objects of other types while generic monoid elements may disallow such a comparison. schmidt.nuigalway.ie /gap/CHAP087.htm   (1667 words)

 MonoidOperation A Monoid Operation is a special sort of Binary Function. The type of the Monoid Operation's first argument and second argument, and also the type returned when the Monoid Operation is returned. Monoid operations are required to be associative, but they are not required to be commutative. www.sgi.com /tech/stl/MonoidOperation.html   (428 words)

 Based on Abstract Algebra   (Site not responding. Last check: 2007-11-06) Monoid comprehensions are based on the notion of monoid homomorphisms from abstract algebra. It maps the zero element of the input monoid to the zero of the output monoid, it replaces the unit function with function f, and it distributes over merge. For example, the homomorphism from the set monoid to the + monoid applies f to every element of the set and adds together the results. ranger.uta.edu /~fegaras/talk98/tsld017.htm   (116 words)

 Tableaux Given the tableau monoid M over the positive integers, and a sequence of sequences of non-negative integers Q, return the tableau with the elements of Q as its rows. Given a tableau monoid M, a sequence of positive integers S which is a partition, and a sequence of non--negative integers C, then return the set of all tableau from M having shape S and content C (see section Properties for definition of content). Given a tableau monoid M, and a sequence of positive integers S which is a partition, return a random standard tableau from M of shape S. If a tableau monoid M is not specified then the monoid over the positive integers is used. www.umich.edu /~gpcc/scs/magma/text1174.htm   (4898 words)

 Introduction   (Site not responding. Last check: 2007-11-06) A rewrite monoid M is a finitely presented monoid in which equality between elements of M, called words or strings, is decidable via a sequence of rewriting equations, called reduction relations, rules, or equations. If a rewrite monoid M is confluent its reduction relations, or more specifically its reduction machine, can be used to reduce words in M to their irreducible normal forms under the given ordering, and so the word problem for M can be efficiently solved. The objects are the rewrite monoids and the morphisms are monoid homomorphisms. www.math.niu.edu /help/math/magmahelp/text224.html   (315 words)

 Character Formulas for q-Rook Monoid Algebras (ResearchIndex)   (Site not responding. Last check: 2007-11-06) Abstract: The q-rook monoid Rn (q) is a semisimple C(q)-algebra that specializes when q # 1toC[Rn ], where Rn is the monoid of n n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. 4 Representations of the rook monoid (context) - Solomon 3 rook monoid (context) - Halverson, the - 2001 citeseer.ist.psu.edu /485816.html   (403 words)

 Words The most important example is the monoid of words generated by the positive integers, though Magma also allows the creation of finitely generated ordered monoids. This plactic monoid is in fact isomorphic to the monoid of tableaux over the same generators. The content of a word is a sequence where the i--th position denotes the number of occurrences of the i--th generator in the word. www.math.lsu.edu /magma/text1359.htm   (1295 words)

 [No title]   (Site not responding. Last check: 2007-11-06) *finitely presented monoid*) is a quotient of a free semigroup (resp. Finitely presented semigroups are obtained by factoring a free semigroup by a set of relations (a generating set for the congruence), ie, a set of pairs of words in the free semigroup. free monoid) are not elements of the finitely presented semigroup (resp. www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/build/fpsemi.msk   (1280 words)

 GAP Manual: 80 Garside and braid monoids and groups Garside monoids are a general class of monoids whose most famous examples are the braid and dual braid monoids. A Garside monoid embeds into its group of fractions, which is called a Garside group (it may be that a Garside group has several distinct Garside structures, as we will see is the case for Braid groups of finite Coxeter groups). is in the group but not the monoid, it is represented in fraction normal form where as a special convention the inverses of the atoms are represented by negating the corresponding integer. www.math.jussieu.fr /~jmichel/htm/CHAP080.htm   (3455 words)

 Conscientia Peccati, Monoid, Stillstand   (Site not responding. Last check: 2007-11-06) Monoid is a term from the area of mathematics: A monoid is a closed half group with a neutral element. The live sound of Monoid is much harder and more organic than on the tape and CDR releases. In 1999 (the date in the booklet is wrong, as the release was delayed) the first true" CD was released: Totalausfall" has more different styles then before and the sound is better thanks to new instruments and recording tools. www.ipsi.fraunhofer.de /~steineba/m01.htm   (305 words)

 Monoids   (Site not responding. Last check: 2007-11-06) A monoid is an algebraic structure that captures many A monoid is an algebraic structure that captures most collection and aggregate types. Formally, a monoid is a pair of an associative function, called the merge function, and a value, which is the left and right identity of the merge function. ranger.uta.edu /~fegaras/talk98/tsld012.htm   (87 words)

 Math Forum: Magic Squares: 'Multiplication' in a new context Let (M,*,1) be a monoid and let P be a subset of M. We say that M is freely generated by P if the unique factorization theorem holds with respect to P, i.e. Thus, a commutative monoid is a monoid (M,*,1) in which the operation * is commutative. If (M,*,1) is a commutative monoid and if M has a subset P such that M is freely generated by P as a commutative monoid, we say that M is a free commutative monoid. www.mathforum.com /alejandre/magic.square/adler/product.html   (961 words)

 Informatica   (Site not responding. Last check: 2007-11-06) Two algebraic structures are employed: Gaussian monoid and a certain module being compatible with a monoid. For both monoid and module, presentation and action level attributes are defined. Monoid action level is defined as monoid element (word) action on module element as an operator. www.vtex.lt /informatica/htm/INFO552.htm   (256 words)

 SEMIGROUPE and MONOID   (Site not responding. Last check: 2007-11-06) MONOID was developed by Steve Linton, Goetz Pfeiffer, Nik Ruskuc, Edmund Robertson. It is a package of GAP functions for transformation monoids and related objects. MONOID provides functions that determine the size of a transformation monoid M, can list the elements of M or decide membership of any transformation of degree n in M. www.inria.fr /actualites/colloques/1998/ALAPEDESMONOID-eng.html   (241 words)

 monoid1 This symbol represents a unary function, whose argument should be a monoid M (for instance constructed by monoid). This symbol represents a unary function, whose argument should be a monoid M. It returns the multiplication map on M. We allow for the map to be n-ary. The unary boolean function whose value is true iff the argument is a commutative monoid. www.openmath.org /cocoon/openmath/cdfiles2/cd/monoid1.html   (930 words)

 Citebase - The quantic monoid and degenerate quantized enveloping algebras We study a monoid associated to complex semisimple Lie algebras, called the quantic monoid. Its monoid ring is shown to be isomorphic to a degenerate quantized enveloping algebra. All these results are proved by a realization in terms of representations of quivers, namely as the monoid of generic extensions of a quiver with automorphism. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0206095   (323 words)

 Conscientia Peccati, Monoid, Stillstand Finished a full demo for the new Monoid CD and am now looking for some label. There will be another Monoid CD this year by DTA records, and I really hope to publish some of the Conscientia Peccati stuff I have waiting since back in 1999 this year  Compilation appearances on Crunch Pod Media, Section 19 and other labels are in progress. There is a new Monoid CD release named “Pain Addiction” planned for September. www.ipsi.fraunhofer.de /~steineba/news.htm   (613 words)

 GAP Share Package "Monoid".   (Site not responding. Last check: 2007-11-06) MONOID is a package of GAP functions for transformation monoids and related objects. MONOID provides functions that determine the size of a transformation monoid M, can list the elements of M or decide membership of any transformation of degree n in M. Moreover, the Green class structure of M can be determined. Relations of degree n can be used to generate a monoid. www.gap-system.org /oldsite/Share/monoid.html   (107 words)

 11. Theory Ensembles Much of mathematics deals with the study of structures such as monoids, fields, metric spaces, or vector spaces, which consist of an underlying set S, some distinguished constants in S, and one or more functions on S. Notice that the theory of a monoid as presented here is suitable for concept formulation and proof in a single monoid. However, there is no way to talk about monoid homomorphisms between different monoids. imps.mcmaster.ca /manual/node16.html   (1273 words)

 OrderedMonoid   (Site not responding. Last check: 2007-11-06) An ordered monoid is a multiplicative monoid together with an ordering of its elements. The reason for making a separate class for ordered monoids is that monoid rings can be implemented more efficiently for them - an element of the monoid ring can be stored as a sorted list, each element of which is a pair consisting of an element of the monoid and a coefficient. A free commutative ordered monoid can be created with monoid. www.math.umn.edu /systems_guide/Macaulay2/html/0315.html   (108 words)

 Monoid   (Site not responding. Last check: 2007-11-06) Note that a monoid satisfies all the axioms of a group with the exception of having inverses. A homomorphism between two monoids (M, *) and (M′, @) is a function f : M → M′ such that Note that not every magma homomorphism is a monoid homomorphism since it may not preserve the identity. www.yotor.com /wiki/en/mo/Monoid.htm   (1067 words)

 MATHS: Monoids Monoid have much of the structure normally taught as part of group theory. What is the smallest monoid with unit 1, operation and element x, such that x x x=1? A monoid is an abstract model of most situations in which a set of actions tend to change objects. www.csci.csusb.edu /dick/maths/math_33_Monoids.html   (766 words)

 All Countable Monoids Embed In The Monoid Of The Infinite Random Graph (ResearchIndex)   (Site not responding. Last check: 2007-11-06) Abstract: We prove that the endomorphism monoid of the infinite random graph R contains as a submonoid an isomorphic copy of each countable monoid. As a corollary, the monoid of R does not satisfy any non-trivial semigroup identity. We also prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of the monoid of R. (Update) citeseer.ist.psu.edu /580877.html   (367 words)

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