Uniform interpolation and compact congruences

All seven intermediate logics admitting Craig interpolation also admit uniform interpolation; however, although the modal logic K admits both properties, its extension S4 admits only Craig interpolation and not uniform interpolation (see [1] for details and references). Uniform interpolation for a logic may be viewed as a weaker form of quantifier elimination. This idea is exploited in the monograph [1] of Ghilardi and Zawadowski to show that under certain conditions, satisfied in particular by IPC and K, uniform interpolation for a logic implies the existence of a model completion for a corresponding variety (equational class) of algebras. In this work, we investigate uniform interpolation in a universal algebraic setting. Following the category-theoretic work in [1], we obtain algebraic characterizations of the property of existence of left and right uniform interpolants. Moreover, we identify, among varieties of algebras corresponding to substructural and many-valued logics, several varieties that admit and do not admit these properties.