Quasigroup

In mathematics, a quasigroup is a magma with a binary operation, in the for all and in the equations

And

Each have exactly one solution, that is, the solution exists and is unique.

A quasigroup is to be distinguished from structures in which only the so-called reduction property (see below) is required. There, even the uniqueness of the solutions of these equations is required, but if ever there is a solution. Sometimes it is required that the underlying set is not empty.

A finite magma is exactly then a quasigroup if the link table is a Latin square, ie, if in each row and in each column of the table of occurrences of each element of exactly once.

• 7.1 isotopy and Parastrophie

Examples

Properties

Each quasigroup has the reduction property, ie

• Follows
• Follows

The reason is that the left equations mean and solutions of the equation are (respectively). Because in a quasigroup most one solution to the equation exists, follows or

In other words, the reduction in property means nothing else than that both describe the left-and right multiplication with an element from an injective mapping of into itself or because injectivity and surjectivity for finite sets are identical, the two images are bijective for finite seen. But in the general case (that is, including infinite ) gives us a bijective since the surjectivity of each equation and is guaranteed by the existence of the solution. Since it provides for each image of a left or right multiplication by the element a preimage

The bijectivity of these two figures is a defining characteristic of quasigroups, ie it can be used readily for the alternative definition of quasigroups: A magma is a quasigroup if and only if the induced by the right and left multiplication pictures are bijective in him. The surjectivity guarantees the existence of solutions of the equations (1) and (2), from the injectivity results uniqueness.

Much evidence from the group theory to statements that relate specifically to groups using quite significantly this property. Only use this property (of all properties that arise purely from the group axioms ), so can the statements made are immediately generalized to quasigroups. But many statements that make only slightly stronger conditions can, on special quasigroups - to generalize - which need not be groups.

The link table of a finite quasigroup is a Latin square: A table filled with different symbols in each row and in each column each symbol occurs exactly once. Conversely, every latin square link table a quasigroup. This means that Latin squares and forth herein abstract descriptive definition only two different, fundamentally equal representations of the same mathematical object finite quasigroup.

Parastrophien

One can in a quasigroup two additional links, called Parastrophien, define: For and out is the solution and is the solution of (think of these two as a "quasi - breaks " or left and right breaks "b left by a "and" b right by a " think ). Then obviously applies:

The first two equations describe the solubility of and, and the other two equations, the uniqueness of the solutions. It is a quasigroup so also defined as algebraic structure with three binary operations that satisfy the aforementioned four equations.

If a group, then and If the quasigroup is commutative, then the two exposures, the unique solvability of ( 1) and (2) are equivalent and the links and are inverses of each other.

For an arbitrary quasigroup are and always quasi groups, the latter linkage is explained, by reversing the multiplication. Overall, one can introduce such a quasigroup six quasigroup links that are called to parastroph. Summing up the link as a relation to, for example, for the original link, then you see that the para disasters links are generated by the operation of the 6 permutations in the symmetric group, the table compare the beginning of this section. The six parameters of verses need not be all different from each other. As a result of the web formula can exist for a quasigroup link exactly 1,2,3 or 6 different parameters stanzas. → See the case of a finite quasigroup and Latin square # Parastrophie.

• If an elementary abelian 2- group then all Parastrophien are identical, this is already sufficient that Q is a commutative quasigroup with inverse property, in which each element is the inverse of itself.
• For a commutative quasigroup is left broken and breaking the law are inverses of each other and there are one or three different Parastrophien.

Note that a para stanza a group in general need not be a group, but if and only associative if its inverse is associative. Therefore, the two para disasters links are ( as and ) each group both shortcuts on Q or not each of the two.

Equivalent descriptions of quasigroups

Other alternative definitions such as the definition described under properties of a quasigroup as magma in which induce the left and right multiplication bijective maps. But another, for initially made ​​definition only slightly modified form, can already achieve a slightly different view of quasigroups: A quasigroup is a magma (quantity with tens of inner link ), in which in each equation of the form depending on two elements (from ), imply the existence of third parties ( in ) and uniquely determined. This definition is somewhat redundant, as it leads to the existence and uniqueness of already from the definition of the inner join, but it describes equal and just relations among the elements.

Quasigroup with inverse property

A quasigroup with inverse property (english inverse property IP) is a magma in which there is a unique element for all, so that for all cases:

As the name indicates, a quasigroup with inverse property is a quasigroup, what we want to prove here. We first show that a solution to the equation exists with on and off; for the existence of follows analogously. To this end let Then it follows from the left side of the inverse equation:

Multiplication from the left with are so However, this means bringing a solution of the equation.

The uniqueness of the solution ( and analogously to the solution) follows because only A and C and depends on the assignment

Is unique to each sub-step.

Loop

Has a quasigroup a neutral element, then it is called a loop. Directly follows from the definition of the quasigroup that each element is a left inverse and a right inverse element has in a loop, but - in contrast to the situation in a group - do not need to match (see also inverse element ). The structure of loops is that of groups are very similar.

Moufang loop

A Moufang loop (named after Ruth Moufang ) is a quasigroup in which applies to all and from:

As the name indicates, a Moufang loop is a loop, what we want to prove here. Be a member of and the (uniquely determined ) element with Then for each in:

So after trimming This is a left identity. Now let the (uniquely determined ) element with Then there is left neutral, and

Shortening of results is therefore a right neutral element. So finally we get is a two sided neutral element.

Since in a loop left and right inverse exist, one shows similarly that they are identical in a Moufang loop: to be out and left and right inverse. Then in particular there is quite neutral and then there multiplication from the right:

Shortening of results

Each associative quasigroup is a Moufang loop, and as associative loop thus a group ( as the group axioms are fulfilled obviously). This indicates that the groups are exactly the associative quasigroups (or quasi those groups which groups are also semi- simultaneously).

Applications

Loops occur for example when, in the geometry of synthetic

In both cases, the additive and multiplicative structure, structure coordinates of a loop region. - The second example is a special case of the first, where you can attach differently to the introduction of coordinates in an affine translation plane than in the general case.

→ See Ternärkörper.

Morphisms

Are quasigroups and illustrations, it means a triple Homotopismus if for all

If all three pictures bijective, then that means a Isotopismus and the two quasigroups are called isotopic to each other.

Are the three figures are identical, ie homomorphism. If, in addition bijective, then isomorphism.

Through three bijective self-maps can at any quasigroup a new isotopic quasigroup link are introduced by

Each to isotopic quasigroup is isomorphic to one of the linking structures generated. If the links are the same, is called a Autotopismus of.

• An important application have Isotopismen in geometry, see Isotope ( geometry).
• For finite quasigroups the Isotopismen lead to an equivalence classification of the corresponding Latin squares in Isotopieklassen, see Latin square # equivalence.

Isotopy and Parastrophie

Isotopy and Parastrophie can also coincide: If a quasigroup with inverse property, then applies

Hence, the links break Para verse isotope to the Isotopismus and the breach of law Para verse about the Isotopismus