Modal expansions of ririgs

TL;DR

This paper introduces I-modal ririgs, characterizes their algebraic structure, and develops a corresponding logic with a Hilbert-style calculus, advancing the algebraic and logical understanding of this variety.

Contribution

It provides a new algebraic variety of I-modal ririgs, characterizes their congruence lattice, and formulates an associated logic with a parametrized deduction theorem.

Findings

Characterization of the congruence lattice via I-filters

Axiomatic presentation of the generated variety

Development of a Hilbert-style calculus with deduction theorem

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 on 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 of 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.