Quasi-group

In mathematics , a quasi-group is a magma with a binary link in the for all and in the equations ${\displaystyle Q}$ ${\displaystyle \star \colon Q\times Q\rightarrow Q}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle Q}$

${\displaystyle a\star x=b}$

and

${\displaystyle y\star a=b}$

each have exactly one solution for x and y. That is, a solution exists and is unique.

A quasi-group is to be differentiated from structures in which only the so-called shortening property (see below) is required. There the uniqueness of the solutions of these equations is required, but only if a solution exists at all. Sometimes it is required that the underlying set is not empty.

A finite magma is a quasi-group if and only if the link table is a Latin square , i.e. if every element occurs exactly once in every row and in every column of the table . ${\displaystyle Q}$

Examples

1. Each group is a quasi-group, because it is fulfilled exactly for and exactly for . ${\displaystyle a\star x=b}$ ${\displaystyle x=a^{-1}\star b}$ ${\displaystyle y\star a=b}$ ${\displaystyle y=b\star a^{-1}}$
2. On each vector space over a body of characteristic not equal to 2, a quasi group can be defined by linking introduces. ${\displaystyle x\star y=(x+y)/2}$
3. On the set of points each Steiner triple system can define a quasigroup: It is for the points this block plan defined and is the third point of the unique cluster of and contains. ${\displaystyle x\star x=x}$ ${\displaystyle x\neq y}$ ${\displaystyle z=x\star y}$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle y}$
4. The set of non-zero elements in a finite - dimensional algebra with zero divisors is a quasi-group with regard to multiplication (e.g. the octaves without 0).
5. The only quasi-group of order 2 is the cyclic group . There are five quasi-groups of order 3, only one of which is a group. The smallest real loop (which is not associative) has order 5. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

properties

Each quasi-group has the shortening property , i. H.

• ${\displaystyle a\star c=b\star c}$ follows ${\displaystyle a=b}$
• ${\displaystyle c\star a=c\star b}$ follows ${\displaystyle a=b}$

This is because the equations on the left mean that and are solutions to the equation (and, respectively ). Since there is at most one solution for the equation in a quasi-group, it follows or ${\displaystyle x_{1}=a}$ ${\displaystyle x_{2}=b}$ ${\displaystyle a\star c=x\star c}$ ${\displaystyle c\star a=c\star x}$ ${\displaystyle x_{1}=x_{2}}$ ${\displaystyle a=b.}$

In other words, the reduction in property means nothing more than that, both the left- and right multiplication with an element of an injective mapping describe in itself or because injectivity and surjectivity are identical for finite sets, the two pictures of finite are visible bijective . But also in the general case (i.e. including infinite ) the bijectivity results, since the surjectivity is guaranteed by the existence of the solution of every equation or . Because with that there is an archetype for every image of a left or right multiplication with the element ${\displaystyle a}$ ${\displaystyle Q}$ ${\displaystyle Q}$ ${\displaystyle (x\mapsto a\star x}$ ${\displaystyle x\mapsto x\star a).}$ ${\displaystyle Q}$ ${\displaystyle Q}$ ${\displaystyle x\star a=y}$ ${\displaystyle a\star x=y}$ ${\displaystyle y}$ ${\displaystyle a}$ ${\displaystyle x.}$

The bijectivity of these two mappings is a defining property of the quasi-groups; H. it can easily be used for the alternative definition of quasi-groups: A magma is a quasi-group if and only if the images induced by the right and left multiplication are bijective in it. The surjectivity guarantees the existence of the solutions of equations (1) and (2), the injectivity results in the uniqueness.

Much evidence from group theory, about statements that relate specifically to groups, use this property to a large extent. If you only use this property (of all properties that result purely from the group axioms), the statements made can immediately be generalized to quasi-groups. But also many statements that make only slightly stronger assumptions can be generalized to special quasi-groups - which do not have to be groups.

The link table of a finite quasi-group is a Latin square : a table filled with different symbols, in which every symbol occurs exactly once in every row and in every column. Conversely, every Latin square is a link table of a quasi-group. Thus, Latin squares and the abstract, descriptive definition given here are only two different representations of the same mathematical object, finite quasi-groups , with equal rights in principle . ${\displaystyle n\times n}$ ${\displaystyle n}$

Parastrophies

Parastrophe shortcuts. The 6 statements in the first column are equivalent to each other and to , which defines them. ${\displaystyle a\star b=c}$ ${\displaystyle V_{k}}$

Linking as a relation	permutation	equivalent description	meaning
${\displaystyle (a,b,c)\in V_{1}}$	identity	${\displaystyle a\star b=c}$	original link
${\displaystyle (b,a,c)\in V_{2}}$	(1.2)	${\displaystyle b\operatorname {\bar {\star }} a=c}$	reverse link
${\displaystyle (a,c,b)\in V_{3}}$	(2.3)	${\displaystyle a\operatorname {\backslash } c=b}$	Left break
${\displaystyle (c,b,a)\in V_{4}}$	(1.3)	${\displaystyle c\operatorname {/} b=a}$	Breaking the law
${\displaystyle (b,c,a)\in V_{5}}$	(3.2.1)	no link sign	Left break of the reverse link
${\displaystyle (c,a,b)\in V_{6}}$	(1,2,3)	no link sign	Right break of the reverse link

One can in a quasigroup two more shortcuts, called the one Parastrophien, define: For and of whether the solution and let the solution of (you can these two as "quasi-breaks" or left and right breaks "b left by- a ”and“ b right-through a ”think). Then obviously: ${\displaystyle (Q,\star )}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle Q}$ ${\displaystyle a\operatorname {\backslash } b}$ ${\displaystyle a\star x=b}$ ${\displaystyle b\operatorname {/} a}$ ${\displaystyle y\star a=b}$

${\displaystyle {\begin{aligned}a\star (a\operatorname {\backslash } b)=b\\(b\operatorname {/} a)\star a=b\\a\operatorname {\backslash } (a\star b)=b\\(b\star a)\operatorname {/} a=b\\\end{aligned}}}$

The first two equations express the solvability of and , and the other two equations express the uniqueness of the solutions. A quasi-group can therefore also be defined as an algebraic structure with three binary connections that satisfy the four equations just mentioned. ${\displaystyle a\star x=b}$ ${\displaystyle y\star a=b}$ ${\displaystyle (Q,\star ,\backslash ,\operatorname {/} )}$

If a group, then and if the quasi-group is commutative , then the two requirements for the unambiguous solvability of (1) and (2) are equivalent and the links and are inversions of each other. ${\displaystyle Q}$ ${\displaystyle a\operatorname {\backslash } b=a^{{-}1}\star b}$ ${\displaystyle b\operatorname {/} a=b\star a^{{-}1}.}$ ${\displaystyle \operatorname {/} }$ ${\displaystyle \operatorname {\backslash } }$

For any quasi-group there are also , and always quasi-groups, the latter connection being explained by reversing the multiplication . A total of six quasi-group links can be introduced into a quasi-group , which are called parastrophic to . Summing up the link as a relation on, for example, for the original link, then it will be seen that the para disasters links by the operation of the 6 permutations in the symmetric group of are generated, the table compare at the beginning of this section. The six parastrophes of need not all be different from one another. As a result of the orbit formula , exactly 1, 2, 3 or 6 different parastrophes can exist for a quasi-group link. → See also Latin square # Parastrophy for the case of a finite quasi-group . ${\displaystyle (Q,\star )}$ ${\displaystyle (Q,\operatorname {\backslash } )}$ ${\displaystyle (Q,\operatorname {/} )}$ ${\displaystyle (Q,\operatorname {\bar {\star }} )}$ ${\displaystyle a\operatorname {\bar {\star }} b=b\star a}$ ${\displaystyle (Q,\star )}$ ${\displaystyle (Q,\star )}$ ${\displaystyle (a,b,c)\in V_{1}\Leftrightarrow a\star b=c}$ ${\displaystyle S_{3}}$ ${\displaystyle V_{1}}$ ${\displaystyle (Q,\star )}$

Examples

• If an elementary Abelian 2-group is then all parastrophies are identical, it is already sufficient that Q is a commutative quasi-group with inverse property, in which each element is inverse to itself. ${\displaystyle (Q,\star )}$
• For a commutative quasi-group , left break and right break are inversions of one another and one or three different parastrophies exist. ${\displaystyle V_{1}=V_{2},\;V_{3}=V_{5},\;V_{4}=V_{6}}$

Note that a parastrophe of a group does not generally have to be a group, but is associative if and only if its reverse is associative. Therefore, the two parastrophic links (also and ) are both group links on Q or neither. ${\displaystyle (Q,\star )}$ ${\displaystyle (Q,{\bar {\star }})}$ ${\displaystyle V_{1},V_{2}}$ ${\displaystyle V_{3},V_{6}}$ ${\displaystyle V_{4},V_{5}}$

Equivalent descriptions of quasi-groups

Other alternative definitions are e.g. B. the definition of a quasi-group as magma described under properties , in which the left and right multiplication induce bijective mappings. But also another form, which is only slightly modified from the definition given at the beginning, can achieve a somewhat different view of quasi-groups: A quasi-group is a magma (set with two-digit internal connection), in which in each equation of the form two elements (from ) , condition and uniquely determine the existence of the third party . ${\displaystyle Q}$ ${\displaystyle a\star b=c}$ ${\displaystyle Q}$ ${\displaystyle Q}$This definition is somewhat redundant, since the existence and uniqueness of already result from the definition of the inner connection, but it describes the relationships of the elements with one another more equally and more directly. ${\displaystyle c}$

Quasi-group with inverse property

A quasi-group with inverse property (English inverse property IP ) is a magma in which there is a unique element for all, so that applies to all : ${\displaystyle Q}$ ${\displaystyle a\in Q}$ ${\displaystyle a^{{-}1}}$ ${\displaystyle b\in Q}$

${\displaystyle a^{{-}1}\star (a\star b)=b=(b\star a)\star a^{{-}1}}$ (Inverse property IP).

As the name indicates, a quasi-group with inverse property is a quasi-group, which we want to prove here. We first show that there is a solution to the equation with and off ; the existence of for follows analogously. Let then then follow from the left side of the inverse equation: ${\displaystyle x}$ ${\displaystyle a\star x=c}$ ${\displaystyle a}$ ${\displaystyle c}$ ${\displaystyle Q}$ ${\displaystyle x}$ ${\displaystyle x\star a=c}$ ${\displaystyle w=a\star \left(a^{{-}1}\star c\right).}$

${\displaystyle a^{{-}1}\star w=a^{{-}1}\star \left(a\star \left(a^{{-}1}\star c\right)\right)=a^{{-}1}\star c.}$

Multiplication from left with is so , however, means with which a solution of the equation is. ${\displaystyle \left(a^{{-}1}\right)^{{-}1}}$ ${\displaystyle \left(a^{{-}1}\right)^{{-}1}\star \left(a^{{-}1}\star w\right)=\left(a^{{-}1}\right)^{{-}1}\star \left(a^{{-}1}\star c\right)}$ ${\displaystyle w=c.}$ ${\displaystyle a\star \left(a^{{-}1}\star c\right)=c}$ ${\displaystyle x=a^{{-}1}\star c}$ ${\displaystyle a\star x=c}$

The uniqueness of the solution (and similarly the solution ) follows because only by and dependent and the assignment ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b=a^{{-}1}\star c}$ ${\displaystyle a}$ ${\displaystyle c}$

${\displaystyle a\mapsto a^{{-}1}\mapsto \left(a^{{-}1}\star c\right)\mapsto b}$

is unique in each sub-step.

Loop

If a quasi-group has a neutral element , it is called a loop . Directly from the definition of the quasi-group, it follows that in a loop every element has a left-inverse and a right-inverse element, which - in contrast to the situation in a group - do not have to match (see also inverse element ). The structure of loops is very similar to that of groups.

Moufang loop

A Moufang loop (named after Ruth Moufang ) is a quasi-group in which applies to everyone and from : ${\displaystyle Q}$ ${\displaystyle a,b}$ ${\displaystyle c}$ ${\displaystyle Q}$

${\displaystyle (a\star b)\star (c\star a)=(a\star (b\star c))\star a.}$

This equation is also one of the Moufang identities , namely Identity (M2 ').

As the name indicates, a Moufang loop is a loop, which we want to prove here. Let be an element of and the (uniquely determined) element with Then for each in : ${\displaystyle a}$ ${\displaystyle Q}$ ${\displaystyle e=a\operatorname {\setminus } a}$ ${\displaystyle a\star e=a.}$ ${\displaystyle x}$ ${\displaystyle Q}$

${\displaystyle (x\star a)\star x=(x\star (a\star e))\star x=(x\star a)\star (e\star x),}$

so after shortening it is a left-neutral element. Let now be the (uniquely determined) element with Then it holds that is left-neutral, and ${\displaystyle x=e\star x.}$ ${\displaystyle e}$ ${\displaystyle b=e\operatorname {/} e}$ ${\displaystyle b\star e=e.}$ ${\displaystyle y\star b=e\star (y\star b),}$ ${\displaystyle e}$

${\displaystyle (y\star b)\star e=(e\star (y\star b))\star e=(e\star y)\star (b\star e)=(e\star y)\star e=y\star e.}$

Shortening of results is a legally neutral element. Ultimately what we get is a mutually neutral element. ${\displaystyle e}$ ${\displaystyle y\star b=y,}$ ${\displaystyle b}$ ${\displaystyle e=e\star b=b,}$ ${\displaystyle e}$

Since left and right inverses exist in a loop, they also exist in a Moufang loop. In a Moufang loop the left and right inverse, however, are even identical: to of had and left and right inverse. Then it follows that since is (right) neutral, multiplication from the right with gives: ${\displaystyle a}$ ${\displaystyle Q}$ ${\displaystyle a^{L}}$ ${\displaystyle a^{R}}$ ${\displaystyle e=a\star a^{R}}$ ${\displaystyle e}$ ${\displaystyle a^{L}=a^{L}\star \left(a\star a^{R}\right).}$ ${\displaystyle a^{L}}$

${\displaystyle a^{L}\star a^{L}=\left(a^{L}\star \left(a\star a^{R}\right)\right)\star a^{L}=\underbrace {\left(a^{L}\star a\right)} _{\text{= e}}\star \left(a^{R}\star a^{L}\right)=e\star \left(a^{R}\star a^{L}\right)=a^{R}\star a^{L}.}$

Shortening of results Thus the inverse element of (is unambiguous, since the left inverse or the right inverse is already unique in a loop). ${\displaystyle a^{L}}$ ${\displaystyle a^{L}=a^{R}.}$ ${\displaystyle a^{-1}:=a^{L}=a^{R}}$ ${\displaystyle a}$ ${\displaystyle a^{-1}}$

Every associative quasi-group is a Moufang loop, and as an associative loop it is consequently a group (since the group axioms are then obviously fulfilled). This shows that the groups are exactly the associative quasi-groups (or those quasi-groups that are also semigroups ).

Applications

Loops occur, for example, when in synthetic geometry

1. an affine plane is equipped with a coordinate ternary body as a coordinate area,
2. an affine translation plane is equipped with a coordinate quasi-body as the coordinate area.

In both cases, the additive structure and the multiplicative structure of the coordinate area is a loop. - The second example is a special case of the first, whereby the introduction of coordinates in an affine translation plane can be started differently than in the more general case.

→ See also ternary bodies .

Morphisms

Are quasi groups and figures, it means triple a Homotopismus if for all true ${\displaystyle (Q,\star ),\;(R,\circ )}$ ${\displaystyle \alpha ,\beta ,\gamma :Q\rightarrow R}$ ${\displaystyle (\alpha ,\beta ,\gamma )}$ ${\displaystyle x,y\in Q}$

${\displaystyle \alpha (x)\circ \beta (y)=\gamma (x\star y)}$.

All three pictures bijective, it means a Isotopismus and the two groups are called quasi -isotopic to each other. ${\displaystyle (\alpha ,\beta ,\gamma )}$

If the three mappings are identical , then homomorphism is called . In addition, if it is bijective, then isomorphism . ${\displaystyle \alpha =\beta =\gamma }$ ${\displaystyle \gamma }$ ${\displaystyle \gamma }$

Through three bijective self-mapping , a new isotopic quasi -group link can be introduced on each quasi-group ${\displaystyle \alpha ,\beta ,\gamma :Q\rightarrow Q}$ ${\displaystyle (Q,\star )}$

${\displaystyle x\circ y=\gamma (\alpha ^{-1}(x)\star \beta ^{-1}(y))}$.

Each quasi-group that is isotopic is isomorphic to one of the connecting structures created in this way . If the links are identical , it is called an autotopism of . If the three mappings are also identical , one calls an automorphism . ${\displaystyle (Q,\star )}$ ${\displaystyle (Q,\circ )}$ ${\displaystyle \circ =\star }$ ${\displaystyle (\alpha ,\beta ,\gamma )}$ ${\displaystyle (Q,\star )}$ ${\displaystyle \alpha =\beta =\gamma }$ ${\displaystyle \gamma }$

• Isotopisms have an important application in geometry, see isotopy (geometry) .
• For finite quasi-groups, the isotopisms lead to an equivalence division of the associated Latin squares into isotopic classes , see Latin square equivalence .

Isotopy and parastrophy

Isotopy and parastrophy can also coincide: If a quasi-group with inverse property, then applies ${\displaystyle (Q,\star )}$

${\displaystyle (a^{-1})\backslash b=a\star b\quad }$ and ${\displaystyle a/(b^{-1})=a\star b}$

Thus the left break parastrophe is isotopic to over isotopism and the right break parastrophe over isotopism ${\displaystyle (Q,\backslash )}$ ${\displaystyle (Q,\star )}$ ${\displaystyle (a\mapsto a^{-1},\operatorname {id} ,\operatorname {id} )}$ ${\displaystyle (Q,/)}$ ${\displaystyle (\operatorname {id} ,b\mapsto b^{-1},\operatorname {id} ).}$

literature

• O. Chein, HO Pflugfelder, JDH Smith (Ed.): Quasigroups and Loops: Theory and Application (=  Sigma Series in Pure Mathematics . Volume 8 ). Heldermann Verlag, Berlin 1990, ISBN 3-88538-008-0 .
• Hall, Marshall: The theory of groups . © Macmillan New York, 1959.
• Kurosch, Aleksander Gennadljewitsch: Group theory .

Web links

• Also contains linking tables of quasi-groups of order 3 and a real loop of order 5.
• Hashing method based on quasi-group string transformations
• Overview of the most important cryptographic applications of the quasi-groups (PDF; 443 kB)

Individual references / footnotes

1. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 1 , p. 122 .
2. ↑ ^{a b} The permutation group operates on , the set of all triples of elements of the quasi-group. The original link is mapped to a parastrophic quasi-group link . ${\displaystyle S_{3}}$ ${\displaystyle Q^{3}}$ ${\displaystyle V_{1}\subset Q^{3}}$
3. ↑ ^{a b c d e f} Günther Eisenreich: Lexicon of Algebra . Akademie-Verlag, Berlin 1989, ISBN 3-05-500231-8 .
4. ↑ TE Evans: Chapter I, Varieties of loops and quasigroups in Chein, Pflugfelder, Smith (1990).
5. ↑ Namely the solutions of the equations and ${\displaystyle x\star a=e}$ ${\displaystyle a\star x=e.}$
6. ↑ If you take “from left to right” as the standard for the order in which the connections are calculated and omit the brackets that result in this order, you can better see what is meant: Informally expressed: You can first calculate the two outer pairs and then continue to calculate “normally” (from left to right), or first calculate “the middle” and then continue “normally” - both lead to the same result. ${\displaystyle (a\star b)\star (c\star a)=a\star (b\star c)\star a.}$