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## On Strictly Positive Fragments of Modal Logics with Confluence

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such

as K.2, D.2, D4.2 and S4.2

with unimodal confluence $\Diamond\Box p \to \Box\Diamond p$

as well as the products of modal logics

in the set {K, D, T, D4, S4}, which contain bimodal confluence

$\Diamond_1\Box_2 p \to \Box_2\Diamond_1 p$.

We show that the impact of the unimodal confluence axiom on the

axiomatisation of strictly positive fragments is rather weak.

In the presence of $\top \to \Diamond \top$, it simply disappears and does not contribute to the axiomatisation. Without $\top \to \Diamond \top$

it gives rise to a weaker formula $\Diamond \top \to \Diamond \Diamond \top$. On the other hand, bimodal confluence gives rise to more complicated formulas such

as $\Diamond_1 p \land \Diamond_2^n \top \to \Diamond_1 (p \land \Diamond_2^n \top)$ (which are superfluous in a product if the corresponding factor contains $\top \to \Diamond\top$).%kudinov: formula is replaced by an equivalent one

We also show that bimodal confluence cannot be captured by any

finite set of strictly positive implications.