Unary polynomials in algebras. I (Q798350)

Some aspects of permutational algebras (e.g. unary algebras, vector spaces, monotone continuous functions, extensions with \(\infty\), permutational clones of the 2-element set) are studied. The main result is the following theorem. Let F be a permutational clone on the set A, \(| A|\geq 3\). Suppose that the monoid G of permutations contained in F satisfies the BBC \((=Bounded\) Blocks Condition). Then F is either a unary clone or the clone of algebraic functions of a vector space on A over a finite field. Here BBC is: there exists \(m\in {\mathbb{N}}\) such that for every a,\(b\in A\) there is a block \(B\subseteq A\), \(| B|\leq m\) of G (i.e. for any \(g\in G\) either \(g(B)=B\) or \(g(B)\cap B=\emptyset\) holds) such that a,\(b\in B\). [For Part II cf. the author and \textit{A. Szendrei}, Proc. Klagenfurt Conf. 1982, 273-290 (1983; Zbl 0538.08006).]