Danos and Regnier (1989) introduced the par-switching condition for the multiplicative proof-structures and simplified the sequentialization theorem of Girard (1987) by the use of par-switching. Danos and Regner (1989) also generalized the par-switching to a switching for $n$-ary connectives (an $n$-ary switching, in short) and showed that the expansion property which means that any excluded-middle formula has a correct proof-net in the sense of their $n$-ary switching. They added a remark that the sequentialization theorem does not hold with their switching. Their definition of switching for $n$-ary connectives is a natural generalization of the original switching for the binary connectives. However, there are many other possible definitions of switching for $n$-ary connectives. We give an alternative and natural definition of $n$-ary switching, and we remark that the proof of sequentialization theorem by Olivier Laurent with the par-switching works for our $n$-ary switching; hence that the sequentialization theorem holds for our $n$-ary switching. On the other hand, we remark that the expansion property does not hold with our switching anymore. We point out that no definition of $n$-ary switching satisfies both the sequentialization theorem and the expansion property at the same time except for the purely tensor-based (or purely par-based) connectives.
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