This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.

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This paper presents a cut-elimination procedure for intuitionistic propositional logic in which cut is eliminated directly, without introducing the multiple-cut rule mix, and in which pushing cut above contraction is one of the reduction steps. The presentation of this procedure is preceded by an analysis of Gentzen's mix-elimination procedure, made in the perspective of permuting cut with contraction. We also show that in the absence of implication, pushing cut above contraction does not pose problems for directly eliminating cut.

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