Inner automorphisms of ordered transformation semigroups

Abstract

This paper studies ordered subsemigroups of the posemigroup of partial order-preserving transformations on an ordered set whose automorphisms are all inner. As a first step, we determine the automorphism structure of the ordered transformation monoid of all order-preserving self-maps on a nonempty ordered set. We show that every ordered automorphism is inner and arises from conjugation by an order-preserving bijection of the underlying set. Consequently, the automorphism group coincides with the group of order-preserving bijections, providing a complete description of the automorphism structure in the ordered setting.