Filter-induced entailment relations in paraconsistent Gödel logics

We consider two expansions of G\"odel logic G with two versions of paraconsistent negation. The first one is G₈₍ₕ -- the expansion of G with an involuitive negation ᵢ defined via v (ᵢ) =1-v (). The second one is G² (, -\!<) -- an expansion with a so-called strong negation. This logic utilises two independent valuations on 0, 1 -- v₁ (support of truth or positive support) and v₂ (support of falsity or negative support) that are connected with. Two valuations in G² (, -\!<) can be combined into one valuation v on 0, 1^ -- the twisted product of 0, 1 with itself -- with two components v₁ and v₂. The two logics are closely connected as ᵢ and allow for similar definitions of co-implication -- -\!<: =ᵢ (ᵢ₈) and -\!<: = () -- but do not coincide since the set of values of G² (, -\!<) is not ordered linearly. Our main goal is to study different entailment relations in G₈₍ₕ and G² (, -\!<) that are induced by filters on 0, 1 and 0, 1^, respectively. In particular, we determine the exact number of such relations in both cases, establish whether any of them coincide with the entailment defined via the order on 0, 1 and 0, 1^, and obtain their hierarchy. We also construct reductions of filter-induced entailment relations to the ones defined via the order.