In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction and disjunction `, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms.
Acclavio, M., Maieli, R. (2020). Generalized connectives for multiplicative linear logic. In Leibniz International Proceedings in Informatics, LIPIcs (pp.1-16). Dagstuhl, Germany : Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing [10.4230/LIPIcs.CSL.2020.6].