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In this paper, we consider finite automata with the restriction that whenever the automaton leaves a state it never returns to it. Equivalently we may assume that the states set is partially ordered and the automaton may never move "backwards" to a smaller state.We show that different types of partially-ordered automata characterize different language classes between level 1 and 3/2 of the Straubing-Th茅rien-Hierarchy.In particular, we prove that partially-ordered 2-way DFAs recognize exactly the class UL of unambiguous languages introduced by Sch眉tzenberger in 1976. As shown by Sch眉tzenberger, this class coincides with the class of those languages whose syntactic monoid is in the variety DA, a specific subclass of all "groupfree" (or "aperiodic") semigroups.DAhas turned out to possess a lot of appealing characterizations. Our result adds one more to these: partially-ordered two-way automata recognize exactly those languages whose syntactic monoid is in DA.