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Topic: Commutative monoid

Related Topics

closed monoidal category

Monoid ring

free monoid

monoid algebra

quotient monoid

symmetric monoidal category

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	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)

	Monoid [Definition] (Site not responding. Last check: 2007-10-13)
		Note that a monoid satisfies all the axioms of a group with the exception of having inversesIn mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication.
		A monoid whose operation is commutativeIn mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative.
		Fix a monoid M, and consider its power setIn mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set....
	www.wikimirror.com /Monoid (2582 words)

	monoid (Site not responding. Last check: 2007-10-13)
		In abstract algebra, a branch of mathematics, a 'monoid\' is a set together with a binary operation satisfying certain axioms, detailed below.
		Note that a monoid satisfies all the axioms of a group with the exception of having inverses.
		An monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid).
	www.yourencyclopedia.net /Monoid.html (1003 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: Definition and links.
		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.
	www.encyclopedian.com /mo/Monoid.html (645 words)

	Kids.net.au - Encyclopedia Natural number - (Site not responding. Last check: 2007-10-13)
		This turns the natural numbers (N, +) into a commutative monoid with neutral element 0, the so-called free monoid with one generator.
		This monoid satisfies the cancellation property and can therefore be embedded in a group.
		This turns (N, ) into a commutative* monoid; addition and multiplication are compatible which is expressed in the distribution law: a * (b + c) = ab + ac.
	www.kids.net.au /encyclopedia-wiki/na/Natural_number (662 words)

	[No title]
		The monoid axiom for C thus implies the monoid axiom for R-mod.
		P is a map of monoids and (iii)P has the universal property of the pushout in the category of monoids.
		Since the forgetful functor commutes with filtered colimi* ts, transfinite composites of such pushouts in the monoid* category are still cofibrations in th* e underlying category C. A Cofibrantly generated model categories We need to transfer model category structures to categories of algebras over tr *iples.
	hopf.math.purdue.edu /Schwede-Shipley/last.txt (8006 words)

	Science Fair Projects - Semiring
		The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily an abelian group.
		A commutative semiring is one whose multiplication is commutative.
		R ∪ {−∞} is a commutative, idempotent semiring with max(a,b) serving as semiring addition (identity −∞) and ordinary addition (identity 0) serving as semiring multiplication.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Semiring (914 words)

	PlanetMath: commutative semigroup
		is commutative if the defining binary operation is commutative.
		A monoid which is also a commutative semigroup is called a commutative monoid.
		This is version 1 of commutative semigroup, born on 2002-11-05.
	planetmath.org /encyclopedia/CommutativeSemigroup.html (70 words)

	[No title]
		In particular, given a general monoid A, we can find a cofibrant monoid QA and a weak equivalence and homomorphism QA -!A. Then the model category QA-mod is the homotopy invariant replacement for the category A-mod, which may not ev* *en be a model category.
		The unit S is cofibrant and A is a commutative monoid satisfying the condi- tions of Theorem 2.1; or 2.
		X of monoids is a countable composition of maps Pi -fi!Pi+1, where fi is the pushout in C of a map X^(i+1)^ gi.
	hopf.math.purdue.edu /Hovey/mon-mod.txt (8088 words)

	monoid -- make a monoid (Site not responding. Last check: 2007-10-13)
		The values assigned to these variables (with assign) are the corresponding monoid generators.
		The function baseName may be used to recover the original symbol or indexed variable.
		The class of all monoids created this way is GeneralOrderedMonoid.
	www.msri.org /info/computing/docs/macaulay/2-0.9/0339.html (127 words)

	Citations: Rational sets in commutative monoids - Eilenberg, Schutzenberger (ResearchIndex)
		For a commutative monoid (3, a subset X of 3 is called linear if X aB, 3.1) for some a M and a finite subset B of M. Equivalently, X is called linear if X iab 1.
		This property generally holds in any commutative monoid [25] Rational subsets of N m are classified and characterized in [57] with respect to logical formulae.
		Th.II in [7] Every congruence in a nitely generated commutative monoid M is a rational (or semilinear) subset of M M.
	citeseer.ist.psu.edu /context/342660/0 (3268 words)

	Free semigroup - Enpsychlopedia (Site not responding. Last check: 2007-10-13)
		In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A, with the binary operation of concatenation.
		As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups.
		Given a set A, the free commutative monoid on A is the set of all multisets with elements drawn from A.
	www.grohol.com /psypsych/Free_monoid (527 words)

	[No title]
		A commutative monoid is a strict monoidal category with one object, so in this sense one can talk about a category enriched in a commutative monoid.
		Let the monoid be G. When G is an abelian group, the M and j seem to be determined by elements N_a,b depending on two objects of A. There is more meat in a V-functor.
		If V is the commutative monoid, then a V-enriched category is a set A plus two functions [-,-,-]: A x A x A ---> V [-]: A ---> V satisfying [a,c,d] + [a,b,c] = [a,b,d] + [b,c,d] [a,a,b] + [a] = 0 = [a,b,b] + [b] for all a, b, c, d.
	www.mta.ca /~cat-dist/catlist/1999/comm-monoid (1465 words)

	Natural number - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-13)
		This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator.
		This turns (N, ) into a commutative* monoid with identity element 1; a generator set for this monoid is the set of prime numbers.
		Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.
	encyclopedia.learnthis.info /n/na/natural_number.html (1402 words)

	Monoid - GrokPedia Encyclopedia (Site not responding. Last check: 2007-10-13)
		For instance, it is perfectly possible to have a monoid in which exist two elements a and b and such that
		The axioms required of a monoid operation are exactly those required of a
		Hence, a monoid is essentially the same thing as a category with a single object.
	www.grokpedia.com /en/m/mo/monoid.htm (695 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.
		It is basically the same, but instead of using the notion of monoid, we use the notion of commutative monoid.
		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.
	mathforum.org /alejandre/magic.square/adler/product.html (961 words)

	Upto11.net - Wikipedia Article for Monoid
		Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining ee = e and es = s = se for all s andisin; S.
		The set of all finite strings over some fixed alphabet andSigma; forms a monoid with string concatenation as the operation.
		A homomorphism between two monoids (M, *) and (Mandprime;, @) is a function f : M andrarr; Mandprime; such that
	www.upto11.net /generic_wiki.php?q=monoid (1058 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		A homomorphism of monoids is a function f from a monoid G to a monoid H such that f(1) = 1, and f(ab) = f(a)f(b) for all a and b in G.
		is isomorphic to the additive monoid of nonnegative integers.
		A denial field is a commutative ring that is a field under the denial inequality, and such that 0 is a prime ideal.
	www.math.fau.edu /Richman/AbstrAlg/basicalg.htm (9752 words)

	Atlas: Z-commutativity of Order 4 of the Dihedral Groups by Aleksandar Blazevski (Site not responding. Last check: 2007-10-13)
		The Z-commutativity is a generalization of the ordinary commutativity of a monoid M. This concept is formulated in the monoid algebra Z[M].
		A monoid M is called Z-commutative of order n if all the permutational determinants for all the n-tuples from M are zero in Z[M].
		For any Z-commutative monoid M of some order, the least order n for which M is Z-commutative must be even.
	atlas-conferences.com /cgi-bin/abstract/camb-13 (272 words)

	The Dimensional Ladder
		A commutative monoid is a strict monoidal category with one object.
		A tricategory with one object is a monoidal bicategory.
		Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp.
	math.ucr.edu /home/baez/hda/dimensional_ladder.html (2262 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)

	1 Introduction
		Commutative monoid objects are sheaves of commutative O-algebras.
		A dg-scheme is, roughly speaking, a scheme together with an enrichement of its structural sheaf into commutative differential graded algebras.
		By definition, a dg-scheme is a space obtained by gluing commutative differential graded algebras for the Zariski topology.
	www.mimuw.edu.pl /~jacho/test/HagWord/HagV3se1.html (1572 words)

	1. Introduction
		The usual polynomial ring can be viewed as a monoid ring where the monoid is a finitely generated free commutative monoid.
		Mora studied the class where the free commutative monoid is substituted by a free monoid - the class of finitely generated free monoid rings (compare e.g.
		In this paper we want to show how this can be done for monoid and group rings by giving different notions of reduction and showing how specializing the reduction according to the given group presentation leads to algorithmic solutions for some classes of groups.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/16/paper_html/node1.html (3481 words)

	5. Conclusions (Site not responding. Last check: 2007-10-13)
		Ap95], corresponds directly to the computation of interreduced suffix Gröbner bases in the commutative polynomial ring viewed as a free commutative monoid ring.
		In the context of string rewriting convergent regular presentations for monoids and groups are considered and inductive inference methods have been proposed to detect the patterns.
		Re95] we have shown how the theory of Gröbner bases in monoid and group rings over fields can be lifted to monoid and group rings over reduction rings fulfilling the axioms given in the introduction and some computability conditions, e.g., allowing to compute finite Gröbner bases for ideals in the coefficient domain.
	www.singular.uni-kl.de /Reports_on_ca/16/paper_html/node5.html (1079 words)

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