Oriented Steiner quasigroups.
Given the quasigroups \(Q\), \(K\) and a function \(f\colon Q\times Q\to K\) a new structure of quasigroup, denoted by \(\mathcal Q_f\), can be defined on \(Q\times K\) and it is called the \(f\)-extension of the quasigroup \(K\) by the quasigroup \(Q\) (the function \(f\) is also called factor system of \(\mathcal Q_f\)). A first result of the paper is that if \(K\) is a commutative quasigroup of order 3, the quasigroup \(f\)-extensions of \(K\) with the same factor system are isomorphic for any quasigroup \(Q\).NEWLINENEWLINE Then the author investigates \(f\)-extensions where \(K\) is the group \(\mathbb{Z}_2\) or \(\mathbb{Z}_3\) and \(Q\) is a Steiner quasigroup associated to an oriented Steiner Triple System (briefly STS) \((\mathcal S,T)\) namely an STS where in each block is assigned a cyclic order. This implies that the following orientation function can be defined: \(f^*\colon\mathcal S\times\mathcal S\setminus\{(x,x)\mid x\in\mathcal S\}\to\{\pm 1\}\) with \(f^*(a_1, a_2)=+1\) or \(f^*(a_1,a_2)=-1\) according with the cyclic order of the points of the block determined by \(a_1\) and \(a_2\). When \(K=\mathbb{Z}_2\), the author considers two different \(f\)-extensions obtained by means of the orientation function \(f^*\) and such \(f\)-extensions are called oriented Steiner quasigroups. When \(K=\mathbb{Z}_3\), the factor system is defined in the following way: for all \(x,y\in\mathcal S\), \(f(x,y):=f^*(x,y)\) if \(x\neq y\) and \(f(x,x):=0\) and such \(f\)-extension is called canonical oriented Steiner quasigroup. Some algebraic properties of these classes of quasigroups are investigated such as flexible laws, cross inverse properties, etc. -- In the final section an encryption algorithm based on quasigroup extensions is proposed.
scientific article; zbMATH DE number 6340463