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Topic: Quasigroup

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	PlanetMath: loop and quasigroup
		A loop is a quasigroup which has an identity element.
		What distinguishes a loop from a group is that the former need not satisfy the associative law.
		This is version 1 of loop and quasigroup, born on 2002-09-06.
	planetmath.org /encyclopedia/Quasigroup.html (66 words)

	Quasigroup
		In mathematics, a quasigroup is a set Q with a binary operation, here denoted , with the property that for all a and b in Q there are unique solutions to the equations a x = b and y * a = b.
		A quasigroup group with an identity element is called a loop.
		A Moufang loop is a quasigroup Q in which (a * b) * (c * a) = (a * (b * c)) * a, for all a, b and c in Q.
	www.guajara.com /wiki/en/wikipedia/q/qu/quasigroup.html (535 words)

	Quasigroup
		The multiplication table of a finite quasigroup is called a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
		It follows from the definition of a quasigroup that each element of a loop has both a left inverse and a right inverse.
		More generally, any vector space over a field of characteristic not equal to 2 gives a quasigroup using this operation.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Loop_(mathematics).html (503 words)

	Take a BrainSip (Site not responding. Last check: )
		In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that "division" is always possible.
		A loop is a quasigroup with an identity element.
		The multiplication table of a finite quasigroup is a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
	quasigroup.mestskadoprava.sk (793 words)

	Reference.com/Encyclopedia/Quasigroup
		In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible.
		There are two formal definitions of quasigroup in common use, one of which defines a quasigroup to be a set with one binary operation, and the other which defines a quasigroup to be a set with three binary operations.
		A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that f(xy) = f(x)f(y).
	www.reference.com /browse/wiki/Quasigroup (1261 words)

	Pigeonhole Principle from Interactive Mathematics Miscellany and Puzzles
		Given more than half the elements of a finite quasigroup G, every element of G is the product of two of the given elements.
		Quasigroup G is a groupoid in which, for every a and b, equation ax = b and xa = b have unique solutions which are not necessarily equal unless the associative law holds.
		Let A be the given subset of the quasigroup G. Let x be an element of G. In the multiplication table of G every element appears exactly once in every row and column.
	www.cut-the-knot.org /pigeonhole/quasi.shtml (418 words)

	PlanetMath: medial quasigroup
		A medial quasigroup is a quasigroup such that, for any choice of four elements
		Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)
		This is version 2 of medial quasigroup, born on 2006-12-11, modified 2006-12-11.
	planetmath.org /encyclopedia/MedialQuasigroup.html (101 words)

	Quasigroup
		The definition of a quasigroup Q says that the left and right multiplication operators defined by are bijections from Q to itself.
		A quasigroup or loop homomorphism is a map f : Q → P between two quasigroups such that f(xy) = f(x)f(y).
		A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that for all x, y in Q. A quasigroup homomorphism is just a homo...
	www.experiencefestival.com /quasigroup (1537 words)

	prob003: quasigroup existence
		A quasigroup can be specified by a set and a binary multiplication opertor, * defined over this set.
		Quasigroup existence problems determine the existence or non-existence of quasigroups of a given size with additional properties.
		QG1.m problems are order m quasigroups for which if ab=cd and a 321 b = c 321 d then a=c and b=d.
	www.dcs.st-and.ac.uk /~ianm/CSPLib/prob/prob003/spec.html (226 words)

	No Title
		A quasigroup Q is a groupoid such that the equation x·y=z has a unique solution in Q whenever two of the three elements x, y, z of Q are specified.
		A loop L is a quasigroup with a neutral element.
		Hence the theory of quasigroups and loops is in a sense complementary to the theory of semigroups and monoids.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/loops/doc/loops_manual.html (4661 words)

	Etta Falconer PhD Abstract
		Isotopy is a generalization of the notion of isomorphism, appropriate for the classification of quasigroups and loops.
		We show that a variety of quasigroups is closed under isotopy if and only if it is the class of all quasigroups, all of whose loop isotopes lie in some isotopically closed variety of loops.
		The varieties of quasigroups and loops that are closed under isotopy form isomorphic lattices, Q* and L* with L*, a sublattice of the lattice of loop varieties.
	www.agnesscott.edu /lriddle/women/abstracts/falconer_PhDabstract.htm (371 words)

	FREAKY LINKS
		Examples of quasigroups are, the integers Z with operation + and the nonzero real numbers R with operation · (real multiplication).
		A non associative example of a quasigroup is exhibited by the table below, a child of the triangular grid, as we shall see later.
		The definition of a quasigroup Q forces its operation table to have the property that every element of Q appears exactly once in every row and column of the operation table.
	www.maa.org /editorial/knot/quasi.html (2489 words)

	Constraint Generation via Automated Theory Formation
		When used in finite algebraic domains such as quasigroup theory, given some examples of the algebra, HR invents new concepts and makes and proves theorems using the Otter theorem prover [mccune:ottermanual].
		Quasigroups of every size exist, but for certain specialised classes of quasigroups, there are open questions about the existence of examples.
		The quasigroup constraint imposed an all-different on each row and column, and the constraints imposed by the quasigroup type were implemented via sets of implication constraints.
	www.doc.ic.ac.uk /~sgc/html_papers/colton_cp01.html (1652 words)

	On linear and inverse quasigroups and their applications in code theory / Under examination / Theses / CNAA
		This thesis is devoted to the theory of n-ary and binary quasigroups and their applications in code theory.
		Automorphism groups of n-ary T-quasigroups, n-ary medial quasigroups and some isotopes of binary left distributive quasigroups are researched.
		5.1 On autotopies and automorphisms of n-ary linear quasigroups
	www.cnaa.acad.md /en/thesis/8175 (434 words)

	[No title]
		In this talk I will speak about a quasigroup transformations on strings and their potentials to be used in cryptography.
		To show that, we have developed several cryptographic algorithms: a block cipher, a stream cipher, a hash function with variable length of output that is strongly collision free and a nonlinear pseudo random number generator.
		So, the quasigroup transformations can be used as PRNGs improvers, but also they can be used as randomness improvers of so called "pour sources of randomness".
	www.win.tue.nl /math/eidma/gligoroski.html (329 words)

	Sebastian Freundt -- quasigroup theory (Site not responding. Last check: )
		Originally motivated from combinatorics and web theory quasigroups can be studied from a plain algebraic point of view.
		Quasigroups can be easily described as sets with one binary operation sufficing the cancellation law and divisibilty law.
		Mostly quasigroups, loops and similar structures are used in CodingTheory or CryptoGraphy.
	www.math.tu-berlin.de /~freundt/QuasigroupTheory.html (174 words)

	Citations: Search for idempotent models of quasigroup identities - Zhang (ResearchIndex)
		....particular, are there such quasigroups of order 9, 10, 12, 13, 14, 15 or 16 This was the first problem we investigated, and for no especially good reason we have invested more effort in it than in the others.
		We prefer these quasigroup problems as benchmarks over randomly generated SAT problems for testing constraint solving methods: The problems have fixed solutions; descriptions of the problems are simple and easy to communicate; most important, some cases of the problems remain open, offering....
		We think these quasigroup problems are much better benchmarks than randomly generated SAT problems for testing constraint solving methods: the problems have fixed solutions; descriptions of the problems are simple and easy to communicate; most importantly, some cases of the problems remain open,....
	citeseer.ist.psu.edu /context/254102/0 (1132 words)

	Ivars Peterson's MathTrek -Completing Latin Squares
		A quasigroup is defined in terms of a set, Q, of distinct symbols and a binary operation (called multiplication) involving only the elements of Q.
		The so-called quasigroup completion problem concerns a table that is correctly but only partially filled in.
		Interestingly, the problems that take longest to resolve one way or the other (soluble versus insoluble) lie at the phase transition, where there are enough constraints (filled-in squares) to limit the number of good choices but not enough to show immediately that the case is hopeless.
	www.maa.org /mathland/mathtrek_5_8_00.html (710 words)

	R L Wilson Research Interests
		A loop is a quasigroup with an additional requirement, that it have a (two-sided) identity element.
		A quasigroup, for example is just the case where the operation table is a Latin Square, i.e.
		Viewed in terms of the picture suggested above, the one that gave a relation between a quasigroup and a Latin Square, an isomorphism requires a permutation which works simultaneously on the rows, the columns, and the entries in the table, while an isotopism allows three different permutations.
	www.math.wisc.edu /~wilson/research.html (546 words)

	Comb. Structures Lecture Notes 2
		As was mentioned in the last lecture, one of the modern spurs for interest in Latin squares has come from the investigation of generalizations of the group concept.
		We conclude that each element of the quasigroup occurs exactly once in each row and column, and so the unbordered multiplication table (which is an n×n array) is a latin square.
		Conversely if L is a latin square having a transversal, then at least one of the quasigroups which have L as multiplication table has a complete mapping.
	www-math.cudenver.edu /~wcherowi/courses/m6406/csln2.html (1845 words)

	Aust. Math. Soc. Gazette Vol 24 No 5
		Hence, to prove that a class of quasigroups {\Cal C} is NOT a variety, it suffices to produce a quasigroup in {\Cal C} having a homomorphic image which does NOT belong to {\Cal C}.
		Hence, in order to show that a class of quasigroups {\Cal C} is NOT a variety it suffices to construct a quasigroup belonging to {\Cal C} whose multiplicative part has a homomorphic image onto a combinatorial quasigroup which cannot be the multiplicative part of a quasigroup belonging to {\Cal C}.
		So, let (S, \circ) be a quasigroup of order n satisfying the defining identities and define a collection of 5-cycles C as follows: for each a \neq b \in S, (a, b, a \circ b, b \circ (a \circ b), b \circ a) \in C.
	www.austms.org.au /Publ/Gazette/1997/Jan98/curt2.html (6398 words)

	[loops] 6 Testing properties of quasigroups and loops
		The reader should be aware that although loops are quasigroups, it is often the case in the literature that a property named P can differ for quasigroups and loops.
		For instance, a Steiner loop is not necessarily a Steiner quasigroup.
		is a quasigroup that is both left and right distributive.
	www-history.mcs.st-and.ac.uk /~gap/Manuals/pkg/loops/doc/htm/CHAP006.htm (691 words)

	AMCA: Groups, trees and quasigroup identities by Aleksandar Krapez (Site not responding. Last check: )
		The lattice of Belousov varieties is described in the article: A. Krapez, M. Taylor: Irreducible Belousov equations on quasigroups, Czechoslovak Math.
		All quasigroups satisfying non-Belousov identities are T-quasigroups and therefore isotopic to abelian groups.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/n/q/74.htm (209 words)

	Quadratic quasigroup varieties (Site not responding. Last check: )
		Quadratic functional equations on quasigroups (with A. Krapez), preprint (1986).
		Applying the machinery developed for functional equations, I was able to deduce essential results about quasigroup varieties defined by quadratic equations and will here just state a couple of typical results.
		It has been observed that some quasigroup identities force every quasigroup satisfying them to be isotopic to a group; some identities even force all quasigroups satisfying them to be isotopic to abelian groups (cf.
	www.cse.ogi.edu /~krstic/summary/node9.html (224 words)

	Quasigroup - Wikipedia, the free encyclopedia
		The unique solutions to these equations are often written x = a \ b and y = b / a.
		The nonzero rationals Q* (or the nonzero reals R) with division (÷) form a quasigroup*.
		Moreover, any loop which satisfies any two of the left, right, antiautomorphic, or weak inverse properties satisfies the inverse property.
	en.wikipedia.org /wiki/Quasigroup (1258 words)

	Categorical models and quasigroup homotopies (Site not responding. Last check: )
		In many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms.
		In this paper, the way homotopies may arise in the context of categorical quasigroup model theory is investigated.
		In this context, the algebraic structures are specified by diagram-based logics, such as sketches, and categories of models become functor categories.
	www.maths.tcd.ie /EMIS/journals/TAC/volumes/11/1/11-01abs.html (91 words)

	Quasigroup Completion Problem Applet
		This page contains a Java applet designed to solve the Quasigroup Completion Problem, where a grid of NxN cells must be colored using N colors such that no color appears more than once in any row or column.
		The colors from an empty square's domain are chosen in a deterministic fashion as well (first color in the domain is used).
		Attempts to solve the Quasigroup problem using a randomized depth-first search with forward checking and propagation.
	home.comcast.net /~mry2/java/QuasiGroup.html (1364 words)

	Citebase - Quasigroup of Local-Symmetry Transformations in Constrained Theories (Site not responding. Last check: )
		In the framework of the generalized Hamiltonian formalism by Dirac, the local symmetries of dynamical systems with first- and second-class constraints are investigated in the general case without restrictions on the algebra of constraints.
		It is thereby shown in the general case that the degeneracy of theories with the first- and second-class constraints is due to their invariance under local-symmetry transformations.
		It is also shown in the general case that the action functional and the corresponding Hamiltonian equations of motion are invariant under the same quasigroup of local-symmetry transformations.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9704140 (276 words)

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