Canonical biassociative groupoids

Abstract

In the paper "Free biassociative groupoids", the variety of biassociative groupoids (i.e., groupoids satisfying the condition: every subgroupoid generated by at most two elements is a subsemigroup) is considered and free objects are constructed using a chain of partial biassociative groupoids that satisfy certain properties. The obtained free objects in this variety are not canonical. By a canonical groupoid in a variety V of groupoids we mean a free groupoid (R, ∗) in V with a free basis B such that the carrier R is a subset of the absolutely free groupoid (T_B, ·) with the free basis B and (tu ∈ R ⇒ t, u ∈ R & t∗u = tu). In the present paper, a canonical description of free objects in the variety of biassociative groupoids is obtained.