Opposite-monoid membership preserves word witnesses

Exact statement

For $M=\langle\tau_a:a\in\Sigma\rangle$ , membership of $f$ in $M$ is equivalent to membership of the same underlying element in $M^{\mathrm{op}}$ after reversing a witnessing word; moreover $\mathcal L$ -triviality of $M$ is equivalent to $\mathcal R$ -triviality of $M^{\mathrm{op}}$ .

StatusProved directly from opposite multiplication

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

Context

Opposite multiplication exchanges left and right ideals and reverses the order in which a word multiplies its generators.

Hypotheses and scope

The generator labels and underlying monoid elements are retained in the opposite monoid.

Proof or evidence

If $f=\tau_{a_1}\cdots\tau_{a_k}$ in $M$ , then $f=\tau_{a_k}\star\cdots\star\tau_{a_1}$ in $M^{\mathrm{op}}$ , and conversely. Principal left ideals in $M$ become principal right ideals in $M^{\mathrm{op}}$ .

Opposite multiplication and proof →

Verification notes

The calculation was checked without treating graph reversal, relation transpose, or a new state representation as part of the algebraic duality.

Limitations

Truth of membership is preserved, but a compact input representation need not be preserved.

Open work

Use the duality only together with an explicit size analysis of any representation of the opposite monoid.