Exact kernel criterion for a smaller realization

Exact statement

The marked monoid ⟨s_1,s_2⟩ embeds in (T_m) if and only if there are $t_1,t_2\in T_m$ whose marked word kernel equals the kernel of (s_1,s_2).

StatusProved by constructing the induced embedding in both directions

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

Context

The criterion replaces an embedding of an exponentially large generated monoid by equality of two compactly presented word congruences.

Hypotheses and scope

The target transformations are marked in the same order as the source generators.
The main statement uses unital monoid embeddings.

Proof or evidence

An embedding sends the marked generators to transformations with the same kernel. Conversely, equal kernels make the map η_s(w)↦η_t(w) well-defined, injective, and multiplicative.

Kernel criterion and proof →

Verification notes

Both well-definedness and injectivity were checked; neither is inferred from one-sided kernel inclusion.

Limitations

For non-unital embeddings one must track the idempotent image of the identity or restrict to the associated local monoid.

Open work

Seek an independent check of the non-unital reformulation.