In previous articles, we showed that the category of profinite L-algebras (where L is a normal modal logic with the finite model property) is monadic over Set. Then, we developed sequent calculi for extensions of the language of L with infinitary conjunctions and disjunctions, proving completeness with respect to profinite L-algebras and relating syntactic properties of the calculi with regularity/exactness properties of the category dual to profinite L-algebras. In this paper, we focus on the algebraic perspective: we characterize those extending extensions of S4 whose profinite algebras enjoy such categorical properties.
Keywords: profiniteness, interior algebras, regularity, Barr exactness
2020 MSC: 03B45, 03G05, 18A30, 18A32, 18C35, 18E08
Theory and Applications of Categories, Vol. 45, 2026, No. 51, pp 2171-2213.
Published 2026-10-09.
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