Marked transformation-monoid kernel

Exact statement

For transformations $s_1,s_2\in T_n$ , the marked morphism $\eta_s:\{a,b\}^*\to T_n$ sends $a$ to $s_1$ and $b$ to $s_2$ ; its kernel records exactly which words induce the same element of the generated monoid.

StatusPrecisely stated under the identity-preserving convention

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

Context

The compact input gives generators, not an explicit multiplication table for the potentially exponential generated monoid. The word kernel is the exact behavioral invariant of that marked presentation.

Definitions

(T_n) is the monoid of total functions on ${1,\ldots,n}$ under composition.
Words (u,v) are kernel-equivalent when η_s(u)=η_s(v).

Hypotheses and scope

The two generators are explicitly listed as transformations on n points.
Embeddings preserve the monoid identity unless stated otherwise.

Proof or evidence

The definition separates the small generator input from the abstract monoid it presents and fixes the convention used by the embedding criterion.

Marked monoids and their word kernels →

Verification notes

The source question, unital convention, and non-unital variant were checked separately.

Limitations

A multiplication-table input would define a different complexity problem.

Open work

Retain the marked generators explicitly in any later hardness reduction or minimization algorithm.