Constructions of (2,n)-varieties of groupoids for n = 7, 8, 9


Lidija Goračinova-Ilieva, Smile Markovski


Given positive integer $n>2$, an algebra is said to be a $(2,n)$-algebra if any of its subalgebras generated by two distinct elements has $n$ elements. A variety is called a $(2,n)$-variety if every algebra in that variety is a $(2,n)$-algebra. There are known only $(2,3)$-, $(2,4)$- and $(2,5)$-varieties of groupoids, and there is no $(2,6)$-variety. We present here $(2,n)$-varieties of groupoids for $n=7,8,9$.