The algebraic semantics for the one-variable monadic fragment of the predicate logic $\mathcal{G}\forall_{\sim}$

In this article we characterize the equivalent algebraic semantics for the one-variable monadic fragment of the first-order logic ${\cal G} \forall_{\sim}$ defined by F. Esteva, L. Godo, P. H\'ajek and M. Navara in Residuated fuzzy logics with an involutive negation, Archive for Mathematical Logic 39 (2000). To this end, we first introduce the variety $\mathbb{MG}_{\sim}$ as a certain class of G\"odel 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 $\mathbb{MG}_\sim$ is exactly the desired equivalent algebraic semantics; this is done via a functional representation of finite subdirectly irreducible algebras.