The conjoinability relation in Lambek calculus and linear logic

In 1958 J. Lambek introduced a calculus L of syntactic types and defined an equivalence relation on types: “x ≡ y means that there exists a sequence x = x1, . . . , xn = y (n ≥ 1), such that xi → xi+1 or xi+1 → xi (1 ≤ i < n)”. He pointed out that x ≡ y if and only if there is join z such that x → z and y → z. This paper gives an effective characterization of this equivalence for the Lambek calculi L and LP , and for the multiplicative fragments of Girard’s and Yetter’s linear logics. Moreover, for the non-directed Lambek calculus LP and the multiplicative fragment of Girard’s linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are.