It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.
Joachim Lambek proposed the first noncommutative logic in his 1958 paper Mathematics of Sentence Structure to model the combinatory possibilities of the syntax of natural languages.
The principal novelty of the calculus of structures was its pervasive use of deep inference, which it was argued is necessary for calculi combining commutative and noncommutative operators; this explanation concurs with the difficulty of designing sequent systems for pomset logic that have cut-elimination.
Lutz Straßburger devised a related system, NEL, also in the calculus of structures in which linear logic with the mix rule appears as a subsystem.
Structads are an approach to the semantics of logic that are based upon generalising the notion of sequent along the lines of Joyal's combinatorial species, allowing the treatment of more drastically nonstandard logics than those described above, where, for example, the ',' of the sequent calculus is not associative.