### 5-sparse Steiner triple systems of order $n$ exist for almost all admissible $n$.

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It is well known that given a Steiner triple system one can define a quasigroup operation $\xb7$ upon its base set by assigning $x\xb7x=x$ for all $x$ and $x\xb7y=z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems....

There are four resolvable Steiner triple systems on fifteen elements. Some generalizations of these systems are presented here.

In H. Kiechle's publication ``Theory of K-loops'' [3], the name Kikkawa loops is given to symmetric loops introduced by the author in 1973. This concept started from an analogical imagination of sum of vectors in Euclidean space brought up on a sphere. In 1975, this concept was extended by him to the more general concept of homogeneous loops, and it led us to a non-associative generalization of the theory of Lie groups. In this article, the backstage of finding these concepts will be disclosed from...