Abstract

In this paper, we introduce the variety of |$I$|-modal ririgs. We characterize the congruence lattice of its members by means of |$I$|-filters, and we provide a description of |$I$|-filter generation. We also provide an axiomatic presentation for the variety generated by chains of the subvariety of contractive |$I$|-modal ririgs. Finally, we introduce a Hilbert-style calculus for a logic with |$I$|-modal ririgs as an equivalent algebraic semantics and we prove that such a logic has the parametrized local deduction-detachment theorem.