Density revisited

In this (part survey) paper,we revisit algebraic and proof-theoretic methods developed by Franco Montagna and his co-authors for proving that the chains (totally ordered members) of certain varieties of semilinear residuated lattices embed into dense chains of these varieties, a key step in establishing standard completeness results for fuzzy logics. Such “densifiable” varieties are precisely the varieties that are generated as quasivarieties by their dense chains.By showing that all dense chains satisfy a certain e-cyclicity equation, we give a short proof that the variety of all semilinear residuated lattices is not densifiable (first proved by Wang and Zhao). We then adapt the Jenei–Montagna standard completeness proof for monoidal t-norm logic to show that any variety of integral semilinear residuated lattices axiomatized by additional lattice-ordered monoid equations is densifiable. We also generalize known results to show that certain varieties of cancellative semilinear residuated lattices are densifiable. Finally, we revisit the Metcalfe–Montagna proof-theoretic approach, which establishes densifiability of a variety via the elimination of a density rule for a suitable hypersequent calculus, focussing on the case of commutative semilinear residuated lattices