Two-variable logic on data words extended with guarded predicates from idempotent monoids with linearly ordered two-sided ideals is decidable, shown via equivalence to ordered quasi-normal set automata whose emptiness reduces to ordered multicounter automata.
That is, the labelling by theY-predicates are consistent with the run indicated by theX-predicates
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Set Automata and Limits of Decidability of Two-Variable Logic on Data Words
Two-variable logic on data words extended with guarded predicates from idempotent monoids with linearly ordered two-sided ideals is decidable, shown via equivalence to ordered quasi-normal set automata whose emptiness reduces to ordered multicounter automata.