Quasi-group

A set with a binary operation (usually called multiplication) in which each of the equations $ a x = b $ and $ y a = b $ has a unique solution, for any elements $ a , b $ of the set. A quasi-group with a unit is called a loop.

A quasi-group is a natural generalization of the concept of a group. Quasi-groups arise in various areas of mathematics, for example in the theory of projective planes, non-associative division rings, in a number of questions in combinatorial analysis, etc. The term "quasi-group" was introduced by R. Moufang; it was after her work on non-Desarguesian planes (1935), in which she elucidated the connection of such planes with quasi-groups, that the development of the theory of quasi-groups properly began.

Basic concepts.

The mappings $ R _ {a} : x \rightarrow x a $ and $ L _ {a} : x \rightarrow a x $ are called right and left translations (or displacements) by the element $ a $. In a quasi-group, translations are permutations of the underlying set (cf. Permutation of a set). The subgroup $ G $ of the group of permutations of the set $ Q $ generated by all translations of the quasi-group $ Q ( \cdot ) $ is called the group associated with the quasi-group $ Q ( \cdot ) $. There is a close relation between the structure of $ G $ and that of $ Q ( \cdot ) $.

A homomorphic image of a quasi-group need not, in general, be a quasi-group, but it is a groupoid with division. Corresponding to homomorphisms of a quasi-group onto a quasi-group are the so-called normal congruences. (A congruence (cf. Congruence (in algebra)) $ \theta $ on $ Q ( \cdot ) $ is normal if each of the relations $ a c \theta b c $ and $ c a \theta c b $ implies $ a \theta b $.) In groups all congruences are normal. A sub-quasi-group $ H $ is called normal if there exists a normal congruence $ \theta $ such that $ H $ coincides with one of the congruence classes. There exist congruences on quasi-groups in which two or even all congruence classes with respect to $ \theta $ are sub-quasi-groups.

Associated with every quasi-group operation on a set are two further operations, called the left and right division operations, and denoted respectively by $ / $ and $ \setminus $. They are defined as follows: $ z / y = x $ and $ z \setminus x = y $ if $ x \cdot y = z $. By considering all possible permutations of $ x , y , z $, one can obtain five further quasi-group operations apart from the initial one. Transition from the basic operation to one of the new ones is called parastrophy. Quasi-groups in which all these operations are the same as the basic one are called totally-symmetric or TS-quasi-groups. TS-quasi-groups can also be defined as quasi-groups satisfying the identities $ x y = y x $ and $ x ( x y ) = y $. Idempotent TS-quasi-groups (that is, those satisfying the additional relation $ x ^ {2} = x $) are called Steiner quasi-groups. They are closely related to Steiner triple systems.

One of the most important concepts in the theory of quasi-groups is that of an isotopy. Two quasi-groups $ Q $ and $ Q _ {1} $ are isotopic if there exist three bijections $ \rho , \sigma , \tau : Q \rightarrow Q _ {1} $ such that $ ( xy) ^ \rho = x ^ \sigma y ^ \tau $ for all $ x , y \in Q $. The number of non-isotopic quasi-groups that can be defined on a finite set of cardinality $ n $ is known (1978) only for $ n \leq 8 $.

Basic classes of quasi-groups.

The earliest papers on quasi-groups are related to generalizations of groups in which the requirement of associativity is replaced by weaker conditions, now called the "A" and "B" Sushkevich postulates. A quasi-group satisfies the "A" Sushkevich postulate if the solution $ x $ of the equation $ ( a b ) c = a ( b x ) $ depends only on $ b $ and $ c $, and the "B" postulate if the solution depends only on $ c $. It has been proved that the quasi-groups of these classes are isotopic to groups. In case the solution of such an equation depends on $ a $ and $ c $, the quasi-group is called a left $ F $- quasi-group. A right $ F $- quasi-group is similarly defined via the equation $ ( a b ) c = x ( b c ) $. A quasi-group that is both a left and a right $ F $- quasi-group is called an $ F $- quasi-group. An idempotent $ F $- quasi-group is called a distributive quasi-group and can be defined by the identities:

$$ ( y z ) x = ( y z ) ( z x ) ,\ x ( y z ) = ( x y ) ( x z ) , $$

called the distributive identities. It has been proved that distributive quasi-groups are isotopic to Moufang loops (see Loop). Not all quasi-groups are isotopic to groups. A quasi-group $ Q ( \cdot ) $ is medial if the following identity holds:

$$ ( x y ) ( u v ) = ( x u ) ( y v ) . $$

Every medial quasi-group is isotopic to an Abelian group $ Q ( \cdot ) $ and the isotopy has the form

$$ x \cdot y = \phi x + \psi y + c , $$

where $ \phi , \psi $ are commuting automorphisms of the group and $ c $ is some element of $ Q $( Toyoda's theorem).

Systems of quasi-groups and functional equations.

Suppose that a system of quasi-groups is defined on a set $ Q $. In this case the operations are more conveniently denoted by letters: e.g., instead of $ ab= c $ one writes $ A ( a , b ) = c $. The quasi-group operations on $ Q $ are assumed to be related in some way, most often by identities, called in this case "functional equations" . One can usually solve the problem of finding a system of quasi-groups on $ Q $ satisfying given functional equations. For example, the equation of general associativity

$$ \tag{1 } A _ {1} [ A _ {2} ( x , y ) , z ] \ = A _ {3} [ x , A _ {4} ( y , z ) ] $$

has been solved; namely, it has proved that if four quasi-groups satisfy (1), then they are isotopic to one and the same group $ Q ( \cdot ) $, and the general solution is given by the equalities:

$$ A _ {1} ( x , y ) = \alpha x \cdot \beta y ,\ \ A _ {2} ( x , y ) = \alpha ^ {-} 1 ( \phi x \cdot \psi y ) , $$

$$ A _ {3} ( x , y ) = \phi x \cdot \theta y ,\ A _ {4} ( x , y ) = \theta ^ {-} 1 ( \psi x \cdot \beta y ) , $$

where $ \alpha , \beta , \phi , \psi , \theta $ are arbitrary permutations of $ Q $. The equation of general mediality is similarly solved:

$$ A _ {1} [ A _ {2} ( x , y ) , A _ {3} ( u , v ) ] = \ A _ {4} [ A _ {5} ( x , u ) , A _ {6} ( y , v ) ] . $$

Here all six quasi-groups turn out to be isotopic to one and the same Abelian group.

$ n $-quasi-groups.

A set with an $ n $- ary operation is called an $ n $- quasi-group if each of the equations

$$ a _ {1} \dots a _ {i-} 1 x a _ {i+} 1 \dots a _ {n} = b $$

(where $ b , a _ {1} \dots a _ {n} \in Q $, $ i = 1 \dots n $) has a unique solution. The basic concepts (isotopy, parastrophy, etc.) of the theory of quasi-groups carry over to $ n $- quasi-groups. Each $ n $- quasi-group is isotopic to a certain $ n $- loop (see Loop).

Certain classes of ordinary binary quasi-groups (such as the classes of medial and TS-quasi-groups, etc.) have an analogue in the $ n $- ary case. An $ n $- ary operation $ A $ is reducible if there exist two operations $ B $, $ C $, of arity at least 2, such that

$$ A ( x _ {1} \dots x _ {n} ) = $$

$$ = \ B ( x _ {1} \dots x _ {i-} 1 , C ( x _ {i} \dots x _ {j} ) , x _ {j+} 1 \dots x _ {n} ) $$

(written $ A = B + ^ { i } C $ for short). Otherwise $ A $ is said to be irreducible. An analogue of the theorem on the canonical factorization of positive integers into prime numbers holds for $ n $- quasi-groups.

Combinatorial questions.

The multiplication table of a finite quasi-group, that is, its Cayley table, is known in combinatorics as a Latin square. One of the problems of the combinatorial theory of quasi-groups, finding systems of mutually orthogonal quasi-groups on a given set, is important for the construction of finite projective planes. Two quasi-groups $ A $ and $ B $ defined on a set $ Q $ are orthogonal if the system of equations $ A ( x , y ) = a $, $ B ( x , y ) = b $ has a unique solution for any $ a $ and $ b $ in $ Q $. Orthogonality of finite quasi-groups is equivalent to that of their Latin squares. It has been proved that a system of mutually orthogonal quasi-groups defined on a set of $ n $ elements cannot contain more than $ n - 1 $ quasi-groups.

Another combinatorial concept related to that of a quasi-group is that of a full permutation. A permutation $ \phi $ of a quasi-group $ Q ( \cdot ) $ is said to be full if the mapping $ \phi ^ \prime : x \rightarrow x \phi x $ is also a permutation of $ Q $. Not every quasi-group has a full permutation. A quasi-group admitting a full permutation is called admissible. For an admissible group there exists a quasi-group orthogonal to it and conversely: If a group has a quasi-group orthogonal to it, then the group is admissible. If a finite quasi-group of order $ n $ is admissible, then one can obtain from it a quasi-group of order $ n + 1 $ by a special process (extension).

Algebraic nets.

Quasi-groups have a natural geometric interpretation by means of algebraic nets, also called algebraic webs (see Webs, geometry of). An algebraic net is a set $ N $ consisting of elements of two types, lines and points, with a certain incidence relation between them. (Instead of the word "incident" one can use the expression "passes through" or "lies on" .) Let the set of lines of $ N $ be divided into three classes such that the following axioms hold: 1) two lines of different classes are incident to exactly one common point of $ N $; and 2) each point is incident to exactly one line of each class. Then $ N $ is called a $ 3 $- net. Similarly, a $ k $- net is defined by partitioning into $ k $ classes. The number (cardinality of the set) of lines in each class is the same and is equal to the number (cardinality of the set) of points of any line of the net. It is called the order of the net. Nets can be coordinatized by means of quasi-groups as follows: Suppose one is given a $ 3 $- net $ N $ with sets of lines $ L _ {1} , L _ {2} , L _ {3} $. Let $ Q $ be a set whose cardinality is equal to the order of $ N $. Let one-to-one correspondences between $ Q $ and each of the $ L _ {i} $ be fixed, that is, to each line in $ L _ {i} $ there is given a coordinate in $ Q $. The set $ Q $ becomes a quasi-group (the coordinate quasi-group of the net) if the following operation is defined on it: $ x y = z $ if and only if the common point of the line with coordinate $ x $ in $ L _ {1} $ and the line with coordinate $ y $ in $ L _ {2} $ lies on the line with coordinate $ z $ in $ L _ {3} $. Conversely, each quasi-group is the coordinate quasi-group of some $ 3 $- net. For different one-to-one correspondences between $ Q $ and the $ L _ {i} $ one obtains different, but isotopic, quasi-group structures on $ Q $. Corresponding to each $ 3 $- net in which the order of the classes $ L _ {1} $, $ L _ {2} $ and $ L _ {3} $ is fixed, is the class of all quasi-groups isotopic to one another. A parastrophy of coordinate quasi-groups corresponds to a renumbering of the classes of the net.

There corresponds to each property of a $ 3 $- net an isotopy-invariant (that is, universal) property of the quasi-group. Examples of such properties are closure conditions (cf. Closure condition), the best known of which are the Thomsen, Reidemeister, Bol, and hexagon closure conditions. The Thomsen condition for a coordinate quasi-group means that the relations $ x _ {1} y _ {2} = x _ {2} y _ {1} $ and $ x _ {1} y _ {3} = x _ {3} y _ {1} $ for the elements $ x _ {1} , x _ {2} , x _ {3} $, $ y _ {1} , y _ {2} , y _ {3} $ of it, imply the relation $ x _ {2} y _ {3} = x _ {3} y _ {2} $. The Thomsen condition holds in a quasi-group if and only if the latter is isotopic to an Abelian group. Similar characterizations for a quasi-group are also obtained for the other closure conditions.

$ k $- nets are coordinatized by means of $ k - 2 $ mutually orthogonal quasi-groups.

References

[1]	V.D. Belousov, "Foundations of the theory of quasi-groups and loops" , Moscow (1967) (In Russian)
[2]	V.D. Belousov, "Nonassociative binary systems" Progress in Math. , 5 (1969) pp. 57–76 Itogi Nauk. Algebra. Topol. Geom. 1965 (1967) pp. 63–81
[3]	R.H. Bruck, "A survey of binary systems" , Springer (1971)
[4]	R.H. Bruck, "What is a loop" A.A. Albert (ed.) , Studies in Modern Algebra , Studies in math. , 2 , Math. Assoc. Amer. (1962) pp. 59–99
[5]	J. Aczél, "Quasigroups, nets and nomograms" Adv. Math. , 1 : 3 (1965) pp. 383–450

Comments

Moufang's original paper is [a1].

For recent results and an extensive bibliography on the connection between quasi-groups, Steiner systems and more general combinatorial structures (e.g., "pairwise balanced designs" ), see [a5].

For the connections between quasi-groups, Latin squares and $ 3 $- nets, see [a3], [a4]. The correspondence between quasi-groups and $ 3 $- nets can also be used to study the algebraic structure of a quasi-group from a geometric point of view by considering geometric invariants, such as the group of all collineations, the group of all collineations preserving the classes, or the group of projectivities of the associated $ 3 $- net; see [a2]. For some older literature on results see [a10], [a11].

References

[a1]	R. Moufang, "Zur Struktur von Alternativkörpern" Math. Ann. , 110 (1935) pp. 416–430
[a2]	A. Barlotti, K. Strambach, "The geometry of binary systems" Adv. Math. , 49 (1983) pp. 1–105
[a3]	J. Dénes, A.D. Keedwell, "Latin squares and their applications" , English Univ. Press (1974)
[a4]	G. Pickert, "Projective Ebenen" , Springer (1975)
[a5]	F.E. Bennett, "The spectra of a variety of quasigroups and related combinatorial designs" Discr. Math. , 77 (1989) pp. 29–50
[a6]	V.D. Belousov, "Algebraic nets and quasi-groups" , Stiintsa , Kishinev (1971) (In Russian)
[a7]	V.D. Belousov, "-ary quasi-groups" , Kishinev (1972) (In Russian)
[a8]	V.D. Belousov, "Configurations in algebraic nets" , Kishinev (1979) (In Russian)
[a9]	O. Chein (ed.) H. Pflugfelder (ed.) J.D.H. Smith (ed.) , Theory and applications of quasi-groups and loops , Heldermann (1990)
[a10]	K. Reidemeister, "Grundlagen der Geometrie" , Springer (1930)
[a11]	W. Blaschke, G. Bol, "Geometrie der Gewebe" , Springer (1938)