Oriented Steiner quasigroups

On Steiner Quasigroups of Cardinality

In [12] Quackenbush has expected that there should be subdirectly irreducible Steiner quasigroups (squags), whose proper homomorphic images are entropic (medial). The smallest interesting cardinality for such squags is 21. Using the tripling construction given in [1] we construct all possible nonsimple subdirectly irreducible squags of cardinality 21 (SQ(21)s). Consequently, we may say that the...

Semi-planar Steiner quasigroups of cardinality 3n

It is well known that for each n ≡ 1 or 3 (mod 6) there is a planar Steiner quasigroup (briefly, squag) of cardinality n (Doyen (1969) and Quackenbush (1976)). A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element subsquag (Quackenbush (1976)). In fact, Quakenbush has stated that there should be such semi-planar squags. In this paper, we construct a s...

Quasigroups, right quasigroups and category coverings

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.

On the location of Steiner points in uniformly-oriented Steiner trees

Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

We present a new exact algorithm for the Steiner tree problem in graphs which is based on dynamic programming. Known empirically fast algorithms are primarily based on reductions, heuristics and branching. Our algorithm combines the best known worst-case run time with a fast, often superior, practical performance.