### A deceptive fact about functions

The paper provides a proof of a combinatorial result which pertains to the characterization of the set of equations which are solvable in the composition monoid of all partial functions on an infinite set.

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The paper provides a proof of a combinatorial result which pertains to the characterization of the set of equations which are solvable in the composition monoid of all partial functions on an infinite set.

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

Any amicable pair ϕ, ψ of Sturmian morphisms enables a construction of a ternary morphism η which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in SL±(2,ℕ) and we study incidence matrices associated with the corresponding ternary morphisms η.

We investigate the factor of the groupoid of terms through the largest congruence with a given set among its blocks. The set is supposed to be closed for overterms.