  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 B\"{u}chi
automaton and a weighted $LTL$ formula in these fragments, decide whether the
behavior of the automaton coincides with the semantics of the formula.
