On generating sets of diagonal acts over semigroups of isotone and continuous transformations

The diagonal acts (automata) over semigroups of isotone transformations of a partially ordered set and continuous mappings of a topological space into itself are investigated. For the diagonal right act over the semigroup of continuous selfmappings of a compact, a necessary condition to be cyclic is given. For the diagonal act over a semigroup of isotone selfmappings of the set of natural numbers, the absence of a countable set of generators is proved. The connections between the continuity and the isotonness are studied.