Finite n-tape automata over possibly infinite alphabets: Extending a theorem of Eilenberg et al

Eilenberg, Elgot and Shepherdson showed in 1969, [9], that a relation on finite words over a finite, non-unary alphabet with p letters is definable in the first order logic with p + 2 predicates for the relations equal length, prefix and last letter is a (for each letter a ∈ Σ) if and only if it can be recognized by a finite multitape synchronous automaton, i.e., one whose read heads move simultaneously. They left open the characterization in the case of infinite alphabets and proposed some conjectures concerning them. We solve all problems and sharpen the main theorem of [9].