## Lidija Goračinova-Ilieva, Smile Markovski

*Construction of Mendelsohn designs by using quasigroups of (2,q)-varieties*

Comment.Math.Univ.Carolin. 57,4 (2016) 501-514.**Abstract:**Let $q$ be a positive integer. An algebra is said to have the property $(2,q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A~variety $\mathcal {V}$ of algebras is a variety with the property $(2,q)$ if every member of $\mathcal {V}$ has the property $(2,q)$. Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2,q)$, $20$ such that $w_p=a$ and $w_{p+1}=b$, then for the least number with this property we say that it is the period of the sequence generated by the pair $(a,b)$. Then the sequence can be represented by the cycle $(w_0,w_1,\dots ,w_{p-1})$. The main purpose of this paper is to show that all of the sequences generated by pairs of distinct elements in arbitrary finite algebra of a variety with the property $(2,q)$ have the same periods (we say it is the period of the variety), and they contain unique appearance of each ordered pair of distinct elements. Thus, the cycles with period $p$ obtained by a finite quasigroup of a variety with the property $(2,q)$ are the blocks (all of them of order $p$) of a~Mendelsohn design.

**Keywords:** Mendelsohn design; quasigroup; $(2,q)$-variety; t-design

**DOI:** DOI 10.14712/1213-7243.2015.183

**AMS Subject Classification:** 05E15 20N05

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