Describing weighted safety with weighted LTL over product $\omega$-valuation monoids

We define the notion of $k$-safe infinitary series over idempotent ordered totally generalized product $\omega$-valuation monoids that satisfy specific properties. For each element $k$ of the underlying structure (different from the neutral elements of the additive, and the multiplicative operation) we determine two syntactic fragments of the weighted $LTL$ with the property that the semantics of the formulas in these fragments are $k$ -safe infinitary series. For specific idempotent ordered totally generalized product $\omega$-valuation monoids we provide algorithms that given a weighted Büchi automaton and a weighted $LTL$ formula in these fragments, decide whether the behavior of the automaton coincides with the semantics of the formula.