Fully commutative vector valued groupoids. (English) Zbl 0737.20038

Algebra and logic, Proc. Conf., Sarajevo/Yugosl. 1987, 29-41 (1989).

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[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.


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


Zbl 0717.00012