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Topic: Quotient monoid

	Quotient group - Wikipedia, the free encyclopedia
		The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo).
		A quotient group of a group G is a partition of G which is itself a group under this operation.
		The quotient group G / G is isomorphic to the trivial group (the group with one element), and G / {e} is isomorphic to G.
	en.wikipedia.org /wiki/Quotient_group (1323 words)

	Integer - Wikipedia, the free encyclopedia
		Z is not closed under the operation of division, since the quotient of two integers (e.g.
		The integer q is called the quotient and r is called the remainder, resulting from division of a by b.
		This is the basis for the Euclidean algorithm for computing greatest common divisors.
	en.wikipedia.org /wiki/Integer (958 words)

	Kernel (algebra) - Wikipedia, the free encyclopedia
		The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N).
		This is because monoids are not Mal'cev algebras.
		For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings).
	en.wikipedia.org /wiki/Kernel_(algebra) (1742 words)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		Monoids are endowed with an associative operation and a neutral element.
		Inner automorphisms: Inn(G) is isomorphic to the quotient of G by its center.
		Inn(G) is isomorphic to the quotient of G by its center.
	home.att.net /~numericana/answer/groups.htm (4881 words)

	[ref] 51 Finitely Presented Semigroups and Monoids
		finitely presented monoid) is a quotient of a free semigroup (resp.
		Finitely presented monoids are obtained by factoring a free monoid by a set of relations, i.e.
		The functionality available for finitely presented monoids is essentially the same as that available for finitely presented semigroups, and thus the previous sections apply (with the obvious changes) to finitely presented monoids.
	www.dma.unina.it /gap4manual/ref/CHAP051.htm (1752 words)

	The Quotient Module Command
		The generators of N. The third group play the same role as subgroup generators in the Todd-Coxeter algorithm, and are treated specially, while the first two groups are logically equivalent, forming the relations of A in the variety of free finitely-generated associative algebras.
		However, when the underlying monoid of A is actually an fp-group G, the vector enumeration algorithm can use a more efficient technique to process the relations of G, and can take advantage of the fact that the generators of G are known to be invertible.
		If the underlying monoid of an fp-algebra is in fact an fp-group then it should be presented to Magma as such.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text896.htm (1600 words)

	Re: EXPONENTIAL
		But my choice is already complete : simply take the usual monoid of contexts -but ignore the multiplicity of formulas ?A- ; it is plain that idempotents are exactly the contexts ?\Gamma and that they are all in the fact <<1>>.
		So there is no mistake in my definition, only the apparent conflict between - the essential non-unicity of <> - the unicity of my definition of <> But one should remember that the semantics enables one to make constructions that have no direct meaning in terms of syntax.
		Now, if <> is an arbitrary subset of <>, introduce the congruence generated by the equivalence between p and p^2 for p \in K. Then we discover that -in the quotient monoid- equality p=p^2 is just another way to speak of <>, i.e.
	www.cis.upenn.edu /~bcpierce/types/archives/1996/msg00148.html (335 words)

	[No title]
		Let M be an abelian topological monoid with a filtration as ab* *ove, where each in;m is a cofibration.
		This* * is the free abelian monoid generated by the irreducible subvarieties of codimension q in P(* *C_ n Cq).
		Appendix A. In [LLFM96 ] and [LF99 ] it is shown that Mor(X; C11;)+ is an abelian topol ogical monoid* with a multiplicative action of the linear isometries operad L induced by the j* *oin pairing on algebraic cycles.
	www.math.purdue.edu /research/atopology/CohenR-Lima-Filho/charact.txt (9303 words)

	Vector Enumeration
		Given a finite set X, and a ring S, we can define the free S-algebra A generated by X. This can be seen either as the monoid algebra of the free monoid of words in X, or as all expressions in X and k, combined by addition and multiplication.
		This gives us the general form of a finitely-presented algebra in Magma, as the quotient of the monoid algebra of an fp-monoid, by the two-sided ideal generated by some additional relators.
		The vector enumeration algorithm explicitly reconciles these two descriptions of an R-module, in the case where R is a finitely presented k-algebra for a field k, and M is a finitely presented R-module, which also has finite k-dimension.
	www.math.wisc.edu /help/magma/text606.html (1546 words)

	Words (Site not responding. Last check: 2007-10-29)
		Given a word w from an ordered monoid and some positive integer k, return a sequence of k distinct increasing subsequences of the word w, such that the maximal number of entries from w is used.
		Given a plactic monoid P, and a sequence of elements from the ordered monoid that its based on, return the element of P corresponding to the Knuth equivalence class of w_1...
		Given a plactic monoid P and a tableau t which are both associated with the same ordered monoid, return the element of P which is uniquely associated to t.
	www.math.lsu.edu /magma/text1359.htm (1295 words)

	Citations: emes combinatoires de commutation et r - Cartier, Foata (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		A free, partially commutative monoid (or trace monoid) is associated to an alphabet Sigma and a symmetric, irreflexive commutation relation I Sigma Theta Sigma as the quotient monoid M (Sigma; I) Sigma =fab = ba j (a; b) 2 Ig.
		Consider the monoid freely generating by the letters a 1, a 2, an, b 1, b 2, b n, subject only to the commutation relations a i b i = b i a i.
		This generalizes well known results for free monoids and commutative monoids and is a significant improvement to the previously known square time algorithm.
	citeseer.ist.psu.edu /context/6908/0 (2519 words)

	Relations and Presentations
		A monoid is almost a group, but it doesn't have to have inverses.
		Consider the free monoid on A and B, with the relation A = B. There is no relator A/B, because there is no B inverse.
		In fact the quotient monoid cannot be produced using relators, because the kernel is empty.
	www.mathreference.com /grp-free,relat.html (807 words)

	Specification of a Presentation (Site not responding. Last check: 2007-10-29)
		A semigroup with non-trivial relations is constructed as a quotient of an existing semigroup, possibly a free semigroup.
		Given a generators clause consisting of a list of variables x_1,..., x_r, and a set of relations relations over these generators, first construct the free semigroup F on the generators x_1,..., x_r and then construct the quotient of F corresponding to the ideal of F defined by relations.
		Given a generators clause consisting of a list of variables x_1,..., x_r, and a set of relations relations over these generators, first construct the free monoid F on the generators x_1,..., x_r and then construct the quotient of F corresponding to the ideal of F defined by relations.
	www.umich.edu /~gpcc/scs/magma/text223.htm (444 words)

	[No title]
		Con* siderable care has to be taken in formulating the construction for topological monoids, b ut the outcome clarifies the status of the original colimits when K is flag; flag complexes ar *e precisely those for which the colimit and the homotopy colimit coincide.
		BM is a homotopy homomorp* hism for any well-pointed topological monoid* M. For each m 0 we consider the small categories id(m), which consist of m obj* ects and their identity morphisms; in particular, we use the based versions id?(m), whic *h result from adjoining an initial object ?.
		It is important to establish when the simplicial topological monoids Btmgo(, * a, D) are proper simplicial spaces, in the sense of [25], because we are interested in th* *e homotopy type of their realisations.
	www.math.purdue.edu /research/atopology/Panov-Ray-Vogt/0202081.txt (9710 words)

	GAP Manual: 5 Rings
		This chapter contains sections that describe how to test whether a domain is a ring (see IsRing), and how to find the smallest and the default ring in which a list of elements lies (see Ring and DefaultRing).
		The existence of this division with remainder implies that the Euclidean algorithm can be applied to compute a greatest common divisors of two elements, which in turn implies that R is a unique factorization ring.
		This default function is seldom overlaid, because there is seldom a better way to compute the quotient.
	www.math.jussieu.fr /~jmichel/htm/CHAP005.htm (3345 words)

	No Title
		Credit will not necessarily be given for a correct answer if it is not accompanied by an explanation of the reasoning involved.
		Give the multiplication table of the quotient monoid L/R.
		is a semigroup, monoid 6 and/or a group.
	math.yorku.ca /Courses/9697/Math2320/Oldtests/1989Exam/disfin01.html (262 words)

	Category Theory
		Any monoid (and thus any group) can be seen as a category: in this case the category has only one object, and its morphisms are the elements of the monoid.
		Composition of morphisms corresponds to multiplication of monoid elements.
		That the monoid axioms correspond to the category axioms is easily verified.
	plato.stanford.edu /entries/category-theory (11780 words)

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