Jakaltek language

In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang.

Definition

A Moufang loop is a loop Q that satisfies the following equivalent identities (the binary operation in Q is denoted by juxtaposition):

1. z(x(zy)) = ((zx)z)y
2. x(z(yz)) = ((xz)y)z
3. (zx)(yz) = (z(xy))z
4. (zx)(yz) = z((xy)z)

for all x, y, z in Q. These identities are known as Moufang identities.

Examples

• Any group is an associative loop and therefore a Moufang loop.
• The nonzero octonions form a nonassociative Moufang loop under octonion multiplication.
• The subset of unit norm octonions (forming a 7-sphere in O) is closed under multiplication and therefore forms a Moufang loop.
• The basis octonions and their additive inverses form a finite Moufang loop of order 16.
• The set of invertible split-octonions forms a nonassociative Moufang loop, as does the set of unit norm split-octonions. More generally, the set of invertible elements in any octonion algebra over a field F forms a Moufang loop, as does the subset of unit norm elements.
• The set of all invertible elements in an alternative ring R forms a Moufang loop called the loop of units in R.
• For any field F let M(F) denote the Moufang loop of unit norm elements in the (unique) split-octonion algebra over F. Let Z denote the center of M(F). If the characteristic of F is 2 then Z = {e}, otherwise Z = {±e}. The Paige loop over F is the loop M*(F) = M(F)/Z. Paige loops are nonassociative simple Moufang loops. All finite nonassociative simple Moufang loops are Paige loops over finite fields. The smallest Paige loop M*(2) has order 120.
• A large class of nonassociative Moufang loops can be constructed as follows. Let G be an arbitrary group. Define a new element u not in G and let M(G,2) = G ∪ (G u). The product in M(G,2) is given by the usual product of elements in G together with

  $(gu)h=(g{h}^{-1})u$

  $g(hu)=(hg)u$

  $(gu)(hu)=h^{-1}g.$

It follows that $u^2=1$ and $ug=g^{-1}u$. With the above product M(G,2) is a Moufang loop. It is associative if and only if G is abelian.

• The smallest nonassociative Moufang loop is M(S₃,2) which has order 12.
• Richard A. Parker constructed a Moufang loop of order 2¹³, which was used by Conway in his construction of the monster group. Parker's loop has a center of order 2 with elements denoted by 1, −1, and the quotient by the center is an elementary abelian group of order 2¹², identified with the binary Golay code. The loop is then defined up to isomorphism by the equations

  A² = (−1)^|A|/4

  BA = (−1)^|A∩B|/2AB

  A(BC)= (−1)^|A∩B∩C|(AB)C

where |A| is the number of elements of the code word A, and so on. For more details see Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England.

Properties

Associativity

Moufang loops differ from groups in that they need not be associative. A Moufang loop that is associative is a group. The Moufang identities may be viewed as weaker forms of associativity.

By setting various elements to the identity, the Moufang identities imply

• x(xy) = (xx)y   left alternative identity
• (xy)y = x(yy)   right alternative identity
• x(yx) = (xy)x   flexible identity.

Moufang's theorem states that when three elements x, y, and z in a Moufang loop obey the associative law: (xy)z = x(yz) then they generate an associative subloop; that is, a group. A corollary of this is that all Moufang loops are di-associative (i.e. the subloop generated by any two elements of a Moufang loop is associative and therefore a group). In particular, Moufang loops are power associative, so that exponents xⁿ are well-defined. When working with Moufang loops, it is common to drop the parenthesis in expressions with only two distinct elements. For example, the Moufang identities may be written unambiguously as

1. z(x(zy)) = (zxz)y
2. ((xz)y)z = x(zyz)
3. (zx)(yz) = z(xy)z.

Left and right multiplication

The Moufang identities can be written in terms of the left and right multiplication operators on Q. The first two identities state that

• $L_z{L}_x{L}_z(y)=L_{zxz}(y)$
• $R_z{R}_y{R}_z(x)=R_{zyz}(x)$

while the third identity says

• $L_z(x)R_z(y)=B_z(xy)$

for all $x,y,z$ in $Q$. Here $B_z=L_z{R}_z=R_z{L}_z$ is bimultiplication by $z$. The third Moufang identity is therefore equivalent to the statement that the triple $(L_z,R_z,B_z)$ is an autotopy of $Q$ for all $z$ in $Q$.

Inverse properties

All Moufang loops have the inverse property, which means that each element x has a two-sided inverse x⁻¹ which satisfies the identities:

$x^{-1}(xy)=y=(yx)x^{-1}$

for all x and y. It follows that $(xy{)}^{-1}=y^{-1}{x}^{-1}$ and $x(yz)=e$ if and only if $(xy)z=e$.

Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop if and only if every loop isotope of Q has the inverse property. If follows that every loop isotope of a Moufang loop is a Moufang loop.

One can use inverses to rewrite the left and right Moufang identities in a more useful form:

• $(xy)z=(x{z}^{-1})(zyz)$
• $x(yz)=(xyx)(x^{-1}z).$

Lagrange property

A finite loop Q is said to have the Lagrange property if the order of every subloop of Q divides the order of Q. Lagrange's theorem in group theory states that every finite group has the Lagrange property. It was an open question for many years whether or not finite Moufang loops had Lagrange property. The question was finally resolved by Alexander Grishkov and Andrei Zavarnitsine, and independently by Stephen Gagola III and Jonathan Hall, in 2003: Every finite Moufang loop does have the Lagrange property. More results for the theory of finite groups have been generalized to Moufang loops by Stephen Gagola III in recent years.

Moufang quasigroups

Any quasigroup satisfying one of the Moufang identities must, in fact, have an identity element and therefore be a Moufang loop. We give a proof here for the third identity:

Let a be any element of Q, and let e be the unique element such that ae = a. Then for any x in Q, (xa)x = (x(ae))x = (xa)(ex). Cancelling gives x = ex so that e is a left identity element. Now let f be the element such that fe = e. Then (yf)e = (e(yf))e = (ey)(fe) = (ey)e = ye. Cancelling gives yf = y, so f is a right identity element. Lastly, e = ef = f, so e is a two-sided identity element.

The proofs for the first two identities are somewhat more difficult (Kunen 1996).

Open problems

Phillips' problem is an open problem in the theory presented by J. D. Phillips at Loops '03 in Prague. It asks whether there exists a finite Moufang loop of odd order with a trivial nucleus.

Recall that the nucleus of a loop (or more generally a quasigroup) is the set of x such that $x(yz)=(xy)z$, $y(xz)=(yx)z$ and $y(zx)=(yz)x$ hold for all $y,z$ in the loop.

See also: Problems in loop theory and quasigroup theory

See also

• Bol loop
• Gyrogroup

References

• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/
• Edgar G. Goodaire, Sean May, and Maitreyi Raman (1999) The Moufang loops of order less than 64, Nova Science Publishers. ISBN 0-444-82438-3
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• K. Kunen, Moufang quasigroups, Journal of Algebra 183 (1996) 231-234.
• R. Moufang, Zur Struktur von Alternativkörpern, Math. Ann. 110 (1935) 416–430
• Jonathan D. H. Smith and Anna B. Romanowska (1999) Post-Modern Algebra, Wiley-Interscience. ISBN 0-471-12738-8.

External links

• LOOPS package for GAP This package has a library containing all nonassociative Moufang loops of orders up to and including 81.
• Template:Planetmath reference