Algebra and proof theory traditionally represent two distinct approaches within logic: the former is concerned with semantic meaning and structures, the latter with syntactic and algorithmic aspects. In many intriguing cases, however, methods from one field are essential to obtaining proofs in the other. In particular, proof-theoretic techniques involving substructural logics have been used to establish important results for their algebraic counterparts, the so called residuated lattices, and vice versa. The purpose of my talk is to survey some of these developments. The talk also emphasizes the central role of lattice-ordered groups in the study of algebras of logic.
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