A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X   and any relation ρρ on X  , let T(X,ρ)T(X,ρ) be the semigroup of all maps a:X→Xa:X→X that preserve ρρ. Let S(X)S(X) be the symmetric group on X  . If ρρ is reflexive, the group of automorphisms of T(X,ρ)T(X,ρ) is isomorphic to NS(X)(T(X,ρ))NS(X)(T(X,ρ)), the normalizer of T(X,ρ)T(X,ρ) in S(X)S(X), that is, the group of permutations on X   that preserve T(X,ρ)T(X,ρ) under conjugation. The elements of NS(X)(T(X,ρ))NS(X)(T(X,ρ)) have been described for the class of so-called dense relations ρρ. The paper is dedicated to applications of this result.