Journal of the
Korean Mathematical Society JKMS

In ring theory it is well-known that a ring $R$ is (von Neumann) regular if and only if all right $R$-modules are flat. But the analogous statement for this result does not hold for a monoid $S$. Hence, in sense of $S$-acts, Liu ([10]) showed that, as a weak analogue of this result, a monoid $S$ is regular if and only if all left $S$-acts satisfying condition $(E)$ ([6]) are flat. Moreover, Bulmann-Fleming ([6]) showed that $x$ is a regular element of a monoid $S$ iff the cyclic right $S$-act $S/\rho(x,x^2)$ is flat. In this paper, we show that the analogue of this result can be held for automata and then characterize regular semigroups by flat automata.

Keywords: flat automaton, cyclic automaton, principal right congruence, regular semigroup

MSC numbers: 20M17