## Fully commutative vector valued groupoids.(English)Zbl 0737.20038

Algebra and logic, Proc. Conf., Sarajevo/Yugosl. 1987, 29-41 (1989).
[For the entire collection see Zbl 0717.00012.]
Let $$Q$$ be a nonempty set, $$m, n$$ positive integers and $$A$$ a mapping of $$Q^ n$$ into $$Q^ m$$. Then $$(Q,A)$$ is said to be an $$(n,m)$$-groupoid or a vector-valued groupoid. Let $$Q$$ be an arbitrary set. In the $$j$$-th Cartesian power $$Q^ j$$ of $$Q$$ a relation $$\sim$$ is defined as follows: $$(x^ j_ 1)\sim(y^ j_ 1)$$ iff there exists a permutation $$\alpha\in\{1,\ldots,j\}!$$, $$j\in N$$, such that $$(y^ j_ 1)=(x_{\alpha 1},\ldots,x_{\alpha j})$$. The set $$Q^ j/\sim$$ is denoted by $$Q^{(j)}$$, and the elements of that set we shall denote as $$(a_ 1,\ldots,a_ j)=a^ j_ 1$$ for each number $$j$$. Every mapping $$A: Q^{(n)}\to Q^{(m)}$$ is called a fully commutative $$(n,m)$$- operation and $$(Q,A)$$ is called a fully commutative $$(n,m)$$-groupoid. A description of the free generated fully commutative $$(n,m)$$-groupoid is given and a result different from the usual algebras is obtained here. Namely, if $$(Q,A)$$ is a free fully commutative $$(n,m)$$-groupoid ($$m\geq2$$) with a basis $$B$$, then the identity mapping on $$B$$ can be extended to infinitely many automorphisms on $$(Q,A)$$. The notion of fully commutative $$(n,m)$$-quasigroups (shortly: f.c.q.) is discussed and a description of the free f.c.q. by using the notion of partial f.c.q. is given. Finally, finite f.c.q. are considered and some examples of finite f.c.q. are given.

### MSC:

 20N15 $$n$$-ary systems $$(n\ge 3)$$

Zbl 0717.00012