System BV is an extension of multiplicative linear logic with a non-commutative self-dual operator. In this paper, we present systems equivalent to system BV where equalities for unit are oriented from left to right and new structural rules are introduced to preserve completeness. While the first system allows units to appear in the structures, the second system makes it possible to completely remove the units from the language of BV by proving the normal forms of the structures that are provable in BV. The resulting systems provide a better performance in automated proof search by disabling redundant applications of inference rules due to the unit. As evidence, we provide a comparison of the performance of these systems in a Maude implementation.
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