Book contents
- Frontmatter
- Contents
- Preface
- Restricted colorings of graphs
- Polynomials in finite geometries and combinatorics
- Models of random partial orders
- Applications of submodular functions
- Weighted quasigroups
- Graphs with projective subconstituents which contain short cycles
- On circuit covers, circuit decompositions and Euler tours of graphs
- Slicing the hypercube
- Combinatorial designs and cryptography
Weighted quasigroups
Published online by Cambridge University Press: 16 March 2010
- Frontmatter
- Contents
- Preface
- Restricted colorings of graphs
- Polynomials in finite geometries and combinatorics
- Models of random partial orders
- Applications of submodular functions
- Weighted quasigroups
- Graphs with projective subconstituents which contain short cycles
- On circuit covers, circuit decompositions and Euler tours of graphs
- Slicing the hypercube
- Combinatorial designs and cryptography
Summary
Abstract. We introduce the concept of a weighted quasigroup. We show that, corresponding to any weighted quasigroup, there is a quasigroup from which it can be obtained in a certain natural way, which we term amalgamation. Not all commutative weighted quasigroups can be obtained from commutative quasigroups by amalgamation; however, given a commutative weighted quasigroup whose weights are all even, we give a necessary and sufficient condition for the existence of a commutative quasigroup from which it can be obtained by amalgamation. We also discuss conjugates of weighted quasigroups.
We also introduce the concept of a simplex zeroid, and relate this concept to that of a weighted quasigroup.
Weighted quasigroups.
Suppose that we have a finite set S with a closed binary operation. If a, b ε S, we shall denote the result of this binary operation acting on a and b by ab. If the binary operation has the two properties
(i) for each a, b ε S, the equation ax = b is uniquely solvable for x, and
(ii) for each a, b ε S the equation ya = b is uniquely solvable for y,
then S is a quasigroup. It is well-known, and easy to see, that S is a quasigroup if and only if its multiplication table is a latin square. The properties (i) and (ii) amount to the assertion that, in the multiplication table, each element of S occurs exactly once in each row and exactly once in each column.
- Type
- Chapter
- Information
- Surveys in Combinatorics, 1993 , pp. 137 - 172Publisher: Cambridge University PressPrint publication year: 1993
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