CANONICAL BIASSOCIATIVE GROUPOIDS

Biljana Janeva, Snezana Ilic, Vesna Celakoska-Jordanova

Abstract: In the paper \emph{Free biassociative groupoids}, the variety of biassociative groupoids (i.e., groupoids that satisfy 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 \textit{canonical groupoid} in a variety $\mathcal{V}$ of groupoids we mean a free groupoid $(R,*)$ in $\mathcal{V}$ with a free basis $B$ such that the carrier $R$ is a subset of the absolutely free groupoid $(T_B,\cdot)$ with the free basis $B$ and $(tu\in R \Rightarrow t,u\in R  &  t*u=tu)$. In the present paper, a canonical description of free objects in the variety of biassociative groupoids is obtained.

Keywords: Groupoid, subgroupoid generated by two elements, subsemigroup, free groupoid, canonical groupoid

Classification (MSC2000): 08B20; 03C05

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