A characterization of subdirectly irreducible acts

The subdirectly irreducible acts (automata) over semigroups are investigated. In 1974, E. N. Roiz proved that such acts have at most two zeros. Here, we characterize subdirectly irreducible acts with two zeros and reduce the characterization of an act with one zero or without zeros to the structure of its least non-trivial subact. We fully characterize the subdirectly irreducible acts over rectangular bands. As the corollaries we have a characterization of subdirectly irreducible acts over right zero semigroups and the Moghaddassi's result about acts over left zero semigroups.