In this paper, we introduce the notions of extended filter and n-fold strongly integral filter (resp., pseudo-residuated lattice). We give the characterizations of n-fold filters by a extended filter. We construct a new logical system of a pseudo-residuated logic. Afterwards, we show that the classes of n-fold strongly integral (resp., boolean, implicative, fantastic, involutive, strong) pseudo-residuated lattices are subvarieties of the variety of all pseudo-residuated lattices. Finally, we give logics which have the above varieties as models and the connections among them.
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