A characterization of full function algebras (Q1097289)

Let n be an arbitrary positive integer. For every set M the algebra \((M^{M^ n},(f_ 1,...,f_{n+1})\mapsto ((x_ 1,...,x_ n)\mapsto f_ 1(f_ 2(x_ 1,...,x_ n),...,f_{n+1}(x_ 1,...,x_ n))))\) of type \(n+1\) is called the full n-place function algebra over M. An algebra (A,f) of type \(n+1\) is called an n-dimensional superassociative system if the law \(f(f(x_ 1,...,x_{n+1}),x_{n+2},...,x_{2n+1})=f(x_ 1,f(x_ 2,x_{n+2},...,x_{2n+1}),...,f(x_{n+1},x_{n+2},...,x_{2n+1}))\) holds in (A,f). \(a\in A\) is called a constant of (A,f) if \(f(a,x_ 1,...,x_ n)=a\) for all \(x_ 1,...,x_ n\in A\). Two algebras (of the same type) are called equivalent if each one can be embedded into the other. It is proved that (for an arbitrary cardinality \(k>1)\) a given simple n-dimensional superassociative system A with exactly k constants is equivalent to the full n-place function algebra over a set of cardinality k iff any simple n-dimensional superassociative system with exactly k constants can be embedded into A.