  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.
