Transversal system
transversal design, transversal scheme, 
-system
A system 
of sets defined for a given collection of 
pairwise-disjoint finite sets 
, each of which has cardinality 
. Namely: a transversal system 
is a system of 
sets 
(blocks or transversals), containing 
elements each and such that:
1) 
; 
; 
;
2) 
for 
.
In a transversal design, any two elements 
and 
, 
, occur together in exactly one block. The existence of a transversal design 
is equivalent to the existence of an orthogonal array 
.
Transversal designs are used in recursive constructions of block designs (cf. Block design).
A set of 
transversals in 
is called parallel if no two of them intersect. If a transversal design 
contains 
(or more) parallel classes, then it is denoted by 
.
Some of the basic properties of transversal systems are:
a) if 
and 
exist, then so does 
;
b) 
exists if and only if 
exists.
References
| [1] | M. Hall, "Combinatorial theory" , Wiley (1986) |
| [2] | H. Hanani, "The existence and construction of balanced incomplete block designs" Ann. Math. Stat. , 32 (1961) pp. 361–386 |
Comments
The finite sets 
making up the design are called point classes or point groups.
The existence of a transversal design 
is equivalent to the existence of 
mutually orthogonal Latin squares of order 
. If a 
exists, and 
, then 
.
See also [a1]–[a3] for the recursive construction of and existence results for transversal designs (and of their generalization "transversal design with holes" , [a2]).
One of the most important results on recursive construction of transversal designs (due to R.M. Wilson [a4], see also [a1]) is as follows:
Let 
be a 
with point classes 
and 
, let 
be a 
-subset of 
and put 
for 
, and 
for every block 
of 
. Assume the existence of 
for 
and of 
for each block 
of 
. Then 
also exists.
Transversals designs (and their dual structures, nets) are also of considerable geometric and algebraic interest. For instance, a 
is equivalent to an affine or projective plane of order 
, and a 
is basically the same as a quasi-group of order 
. Thus, the geometric and algebraic properties of transversal designs (including the study of their automorphism groups) have found considerable interest, cf. [a4].
References
| [a1] | T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986) |
| [a2] | A.E. Brouwer, "Recursive constructions of mutually orthogonal Latin squares" Ann. Discr. Math. , 46 (1991) pp. 149–168 |
| [a3] | D. Jungnickel, "Latin squares, their geometries and their groups: a survey" D. Ray-Chaudhuri (ed.) , Coding Theory and Design Theory , IMA Vol. Math. Appl. , 21 , Springer (1990) pp. 166–225 |
| [a4] | R.M. Wilson, "Concerning the number of mutually orthogonal Latin squares" Discr. Math. , 9 (1974) pp. 181–198 |
| [a5] | H. Hanani, "Balanced incomplete block designs and related designs" Discrete Math. , 11 (1975) pp. 255–369 |
Transversal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transversal_system&oldid=49025