 For reasons of historical precedence, the system of Lambek 1958, with an associative composition operation, is referred to as L; the pure residuation logic NL, i.e., the non-associative version of L, was introduced later in Lambek 1961.  Addition of commutativity turns L and NL into LP and NLP, respectively. LP, Lambek calculus with Permutation, was introduced in van Benthem 1983; in retrospect, this the implication/fusion fragment of Multiplicative Intuitionistic Linear Logic (MILL). The commutative variant of NL has been studied by Kandulski; it has not found significant linguistic applications. For the four systems in the hierarchy, one writes (N)L(P)* for the (non-conservative) extension which allows empty antecedents.