Target isolation is strictly weaker than faithfulness

Exact statement

For the L-trivial monoid $M=\{1,a,0\}$ with $a^2=0$ , the target $f=1$ has a two-state target-isolating poDFA quotient that maps $a$ and $0$ to the same transformation; hence the quotient is nonfaithful while its target fibre is a singleton.

StatusProved by a complete multiplication-table and action check

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

Context

The example proves that seeking a faithful representation can impose unnecessary state cost on a single-target membership instance.

Definitions

$M=\{1,a,0\}$ , with $1$ an identity, $0$ a zero, and $a^2=0$ .
The image contains the identity and the constant-top transformation of a two-state chain.

Proof or evidence

All nine products preserve the map; the three principal left ideals are distinct; the fibre over the image identity is exactly {1}; and a and 0 collapse.

Three-element monoid example →

Verification notes

The preserved exact script checked the multiplication table, L-triviality, homomorphism law, poDFA order, target fibre, and word membership through length 32. A separate review implementation independently rechecked all nine products, the order, and the singleton fibre.

Limitations

The example demonstrates strict weakening, not an asymptotic separation and not a general construction.

Open work

Search for canonical target-isolating congruences that retain poDFA order while collapsing large non-target regions.