Direct reversal does not produce the required poDFA

Exact statement

The minimal two-state DFA for words ending in $a$ has L-trivial transition monoid $\{1,c_0,c_1\}$ , but reversing its transitions yields a nondeterministic automaton with a two-state cycle and a self-looping state having another outgoing transition on the same letter.

StatusProved by an explicit two-state example

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Algebraic passage to an opposite monoid does not mean reversing the transition graph of the supplied DFA.

Definitions

$c_i$ is the constant map to state $i$ .

Proof or evidence

The two DFA states are reachable and distinguished. The monoid's three principal left ideals are distinct. After edge reversal, the preimages of each constant transition create branching, self-loop nondeterminism, and a directed two-cycle.

Two-state reversal counterexample →

Verification notes

Minimality, L-triviality, nondeterminism, the cycle, and failure of the rpoNFA self-loop condition were checked directly.

Limitations

The counterexample refutes one reduction strategy; it does not refute the possibility of another compact dual action or of NP membership.

Open work

Exclude graph reversal and matrix transposition from future dual-representation arguments unless functionality and order are proved.