Jürgen Wedekind ("Classical logics for attribute-value languages", can be googled up) has defined an attribute-value logic as a fragment of predicate logic.
There are no predicates except for identity ($\approx$), no quantifiers and only unary function symbols. The terms are either constants or have the form $f\sigma$ where $f$ is a function symbol and $\sigma$ a term. The atomic formulae have the form $\sigma\approx\tau$ where $\sigma,\tau$ are terms. The logical connectives are $\neg,\land,\lor,\supset,\equiv$.
How could this logic be algebraized? The logic seems quite simple to me (as compared to predicate logic whose algebraic counterpart are cylindric algebras).