The fluted fragment with transitive relations

The fluted fragment is a of first-order logic (without equality) in which, roughly speaking, the order quantification variables coincides with which those appear as arguments predicates. It known that this has finite model property. We consider extensions various numbers transitive relations, well equality predicate. In presence one relation (together equality), property lost; nevertheless, we show satisfiability and problems for extension remain decidable. also corresponding two relations (with or three are undecidable, even two-variable sub-fragment.