In this article we characterize the equivalent algebraic semantics for the one-variable monadic fragment of the first-order logic \({\mathcal G} \forall _{{\sim }}\) defined by Esteva et al. (Arch Math Logic 39:103–124, 2000). To this end, we first introduce the variety \(\mathbb{M}\mathbb{G}_{{\sim }}\) as a certain class of Gödel algebras endowed with two monadic operators and a De Morgan negation. We study its basic properties, determine its subdirectly irreducible members and prove that this variety has the finite embeddabilty property. In particular, we prove that a special subvariety \(\mathbb {CMG}_{{\sim }}\) of \(\mathbb{M}\mathbb{G}_{{\sim }}\) is exactly the desired equivalent algebraic semantics; this is done via a functional representation of finite subdirectly irreducible algebras.
