Subdirectly irreducible medial quandles

ABSTRACT

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.

KEYWORDS:

• Medial quandles
• modes
• quandles
• reductive algebras
• subdirectly irreducible algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 08B26, 15A78
• Secondary 08A30, 03C05, 08A05, 20N02

Acknowledgements

We would like to thank David Stanovský for his valuable comments and fruitful discussion, and Keith Kearnes for giving to us details of Bergman’s construction. We also wish to thank the referee for his/her comment that each SI medial quandle of quasi-affine type is connected which allowed to complete the classification of infinite SI medial quandles.

After revised version of our paper was submitted, Joseph D. Cyr [3] sent to us a direct proof that each SI medial quandle of set type is quasi-reductive (which follows from his more general results). In the paper, it follows from Theorems 4.5 and 9.1.