  $\nabla$-algebra is a natural generalization of Heyting algebra, unifying
many algebraic structures including bounded lattices, Heyting algebras,
temporal Heyting algebras and the algebraic presentation of the dynamic
topological systems. In a series of two papers, we will systematically study
the algebro-topological properties of different varieties of $\nabla$-algebras.
In the present paper, we start with investigating the structure of these
varieties by characterizing their subdirectly irreducible and simple elements.
Then, we prove the closure of these varieties under the Dedekind-MacNeille
completion and provide the canonical construction and the Kripke representation
for $\nabla$-algebras by which we establish the amalgamation property for some
varieties of $\nabla$-algebras. In the sequel of the present paper, we will
complete the study by covering the logics of these varieties and their
corresponding Priestley-Esakia and spectral duality theories.
