York Semigroup: Endomorphism monoids of relational first-order structures

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There exists a large body of literature on automorphism groups and endomorphism monoids on countably infinite first-order structures M; these are analogues of the symmetric group Sym(X) and full transformation monoid End(X) on a set X respectively. There are other natural semigroups of transformations on X that can be generalised to the structural case; the monoid of all injective transformations Mon(X), the symmetric inverse monoid Inv(X) and the partial transformation monoid Part(X) are three cases in point. Moreover, there are examples of interesting endomorphism monoids on M that have no analogue in the case of a set; for instance, there can exist bijective endomorphisms of M that are not automorphisms of M (and hence have no structure-preserving inverse), whereas any bijection from a set X to itself is necessarily invertible.