Modal expansions of ririgs

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.

Automated summary

This paper introduces various $I$-modal ririgs and characterizes the congruence lattice of its members using $I$-filters. It provides a description of $I$-filter generation and presents an axiomatic presentation for the variety generated by chains of the subvariety of contractive $I$-modal ririgs. Finally, it introduces a Hilbert-style calculus for a logic with $I$-modal ririgs as an equivalent algebraic semantics and proves the parametrized local deduction-detachment theorem for such a logic.