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The 'Long Rule' in the Lambek Calculus with Iteration: Undecidability without Meets and Joins

We consider the Lambek calculus extended with positive iteration as a unary connective. The choice of positive iteration, not Kleene star, is dictated by Lambek’s antecedent non-emptiness restriction. Usually iteration is axiomatized either by an inductive schema or by an ω-rule. We consider an intermediate system with a rule which we call the ‘long rule,’ which reduces iteration of A to explicit treatment of powers of A up to the k-th one, and reusing iteration in the form A · A. In the presence of additive disjunction (union), the ‘long rule’ is easily derivable. For the ‘pure’ Lambek calculus without additives this is not the case. For the system with the ‘long rule’ we prove undecidability. We also investigate connections of this system with the standard inductive-style one.