Steiner \(P\)-algebras

Abstract

General algebraic systems are able to formalize problems of different branches of mathematics from the algebraic point of view by establishing the connectivity between them. It has lots of applications in theoretical computer science, secure communications etc. Combinatorial designs play significant role in these areas. Steiner Triple Systems (STS) which are particular case of Balanced Incomplete Block Designs (BIBD) from combinatorics can be regarded as algebraic systems. Steiner quasigroups (Squags) and Steiner loops (Sloops) are two well known algebraic systems which are connected to STS. There is a one-to-one correspondence between STS and finite Squags and finite Sloops. A new algebraic system w.r.to a ternary operation  \(P\) based on a Steiner Triple System introduced in [3].

In this paper the abstraction and the generalization of the properties of the ternary operation defined in [3] has been made. A new class of algebraic systems Steiner \( P\)-algebras has been introduced. The one-to-one correspondence between STS on a linearly ordered set and finite Steiner \(P\)-algebras has been established. Some identities have been proved.