Publication at Faculty of Mathematics and Physics |
2021
Abstract
A binary operation "." which satisfies the identities x . e = x, x . x = e, (x . y) . x = y and x . y = y . x is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order n with centre of order m and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order.
An STS which attains this maximum for a given order is said to be maxi-Pasch. We show that loop factorization preserves the maxi-Pasch property and find that the Steiner loops of all currently known maxi-Pasch Steiner triple systems have centre of maximum possible order.