Stabilized Identities in Finite Transformation Semigroups

Let Xn={1,2,…,n}. Previous work has focused on ordinary semigroup identities and the structural properties of individual transformation monoids. Building on related identity-based work involving one of the present authors, we compare the full transformation semigroup Tn, the order-preserving semigroup On, the order-preserving-or-order-reversing semigroup ODn, the orientation-preserving semigroup OPn, and the anti-cyclic one-line family ORn, treated only as a subset of Tn. For each family S, we determine the least positive exponent ES such that aES is idempotent for every a∈S. This gives zES=z2ES. For n≥2, the exponents for Tn, On, ODn, and OPn are lcm(1,…,n), n−1, 2n−12, and lcm(1,…,n), while E^(ORn)=2n−12 is the subset exponent for ORn. We then study xESyESxES=yESxES. With p=xES and q=yES, it reduces to pqp=qp. This holds exactly when q maps each kernel block of p into a single kernel block of p, and fails exactly when q splits a block. Together with the automatic cases, this test gives a classification of all ordered pairs into automatic, positive, and negative classes.