  Pomset logic introduced by Retor\'e is an extension of linear logic with a
self-dual noncommutative connective. The logic is defined by means of
proof-nets, rather than a sequent calculus. Later a deep inference system BV
was developed with an eye to capturing Pomset logic, but equivalence of system
has not been proven up to now. As for a sequent calculus formulation, it has
not been known for either of these logics, and there are convincing arguments
that such a sequent calculus in the usual sense simply does not exist for them.
In an on-going work on semantics we discovered a system similar to Pomset
logic, where a noncommutative connective is no longer self-dual. Pomset logic
appears as a degeneration, when the class of models is restricted. Motivated by
these semantic considerations, we define in the current work a semicommutative
multiplicative linear logic}, which is multiplicative linear logic extended
with two nonisomorphic noncommutative connectives (not to be confused with very
different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets
and show how this logic degenerates to Pomset logic. However, a more
interesting problem than just finding yet another noncommutative logic is to
find a sequent calculus for this logic. We introduce decorated sequents, which
are sequents equipped with an extra structure of a binary relation of
reachability on formulas. We define a decorated sequent calculus for
semicommutative logic and prove that it is cut-free, sound and complete. This
is adapted to "degenerate" variations, including Pomset logic. Thus, in
particular, we give a variant of sequent calculus formulation for Pomset logic,
which is one of the key results of the paper.
