The notions of pre-local function, semi-local functions and ${alpha}$-local functions with respect to a topology and an ideal are introduced, and several properties are investigated. Also, the concept of $P-mathcal{I}$-open sets and $P-mathcal{I}$-closed sets in ideal topological spaces are discussed. Relations between $mathcal{I}$-open sets and $P-mathcal{I}$-open sets are provided, and several properties related to $P-mathcal{I}$-open sets, pre-local functions, semi-local functions and ${alpha}$-local functions with respect to a topology and an ideal are investigated.