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Natural deduction and coherence for non-symmetric linearly distributive categories
Theory and applications of categories, The Lambek Festschrift, Tome 6 (1999), pp. 105-146.

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In this paper certain proof-theoretic techniques of [BCST] are applied to non-symmetric linearly distributive categories, corresponding to non-commutative negation-free multiplicative linear logic (mLL). First, the correctness criterion for the two-sided proof nets developed in [BCST] is adjusted to work in the non-commutative setting. Second, these proof nets are used to represent morphisms in a (non-symmetric) linearly distributive category; a notion of proof-net equivalence is developed which permits a considerable sharpening of the previous coherence results concerning these categories, including a decision procedure for the equality of maps when there is a certain restriction on the units. In particular a decision procedure is obtained for the equivalence of proofs in non-commutative negation-free mLL without non-logical axioms.
Classification : 03B60, 03F05, 03F07, 03G30, 18D10, 18D15.
Keywords: categorical proof theory, linear logic, monoidal categories. 
@article{TAC_1999_6_a8,
     author = {Robert R. Schneck},
     title = {Natural deduction and coherence 
for non-symmetric linearly distributive categories},
     journal = {Theory and applications of categories},
     pages = {105--146},
     publisher = {mathdoc},
     volume = {6},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_1999_6_a8/}
}
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Robert R. Schneck. Natural deduction and coherence 
for non-symmetric linearly distributive categories. Theory and applications of categories, The Lambek Festschrift, Tome 6 (1999), pp. 105-146. http://geodesic.mathdoc.fr/item/TAC_1999_6_a8/