In this paper, we algebraically study the \(\left\{ \vee ,\rightarrow ,\square ,\Diamond ,\bot ,\top \right\} \) -fragments of the Positive Intuitionistic Modal Logic (i.e., the Positive Modal Logic \(\textbf{PML}\) with an intuitionistic implication), as well as those of the intuitionistic modal logic \(\textbf{FS}\) defined by G. Fischer Servi. We introduce and define the varieties of Positive Modal Hilbert algebras \((\textrm{PMHil}\) -algebras) and Fischer Servi Modal Hilbert algebras \((\textrm{FSMHil}\) -algebras) and we establish spectral-like dualities for these algebras. We define the categories \(\mathsf {\textbf{PSemHil}}\) and \(\mathsf {\textbf{FSSemHil}}\) , whose objects are \(\textrm{PMHil}\) -algebras and \(\textrm{FSMHil}\) -algebras, respectively, and whose morphisms are \(\square \Diamond \) -semi-homomorphisms. By considering \(\square \Diamond \) -homomorphisms instead, we obtain the categories \(\mathsf {\textbf{PHomHil}}\) and \(\mathsf {\textbf{FSHomHil}},\) respectively. Furthermore, we prove that these categories are dually equivalent to categories of \(H_{0}^{\vee }\) -spaces endowed with a special binary relation and certain special continuous maps. The established dualities enable us to characterize the congruences in \(\textrm{PMHil}\) -algebras and \(\textrm{FSMHil}\) -algebras.
