Ideal convergence in modified IFNS and $ \mathcal{L} $-fuzzy normed space

This paper aims to present the concept of $ I $ & $ I^* $ convergence and $ s_p $- $ I $ convergence along with the $ I $ Cauchy criterion in $ \mathcal{L} $-fuzzy normed space (in short $ \mathcal{L} $-FNS). Characterizations of these notions in $ \mathcal{L} $-FNS have been shown in the paper. This paper also presents how these notions are related to each other in $ \mathcal{L} $-FNS. We have also given certain important counter-examples to establish the relationships between them. In addition, we introduce the $ \mathcal{L} $ -fuzzy limit points and $ \mathcal{L} $-fuzzy cluster points of a sequence in $ \mathcal{L} $-FNS.