Algebraic Relativization and Arrow Logic Maarten Marx Abstract: We investigate several weakened versions of first--order logic, and of the logic of binary relations, as provided by representable relation algebras. The most important reason to weaken these two well-known and often-used logics is their complexity: the theory of both systems is undecidable. These logics are not only used in areas where this complexity is needed, as in mathematics, but also in other disciplines (computer science, linguistics) which deal with simpler (i.e., decidable) questions. For this reason it is desirable to develop new versions of these logics, which are close to the original with respect to expressive power and semantics, but behave better computationally. We change first--order logic such that, given a model $M=$ the set of assignments for $M$ is just a subset of ${^\omega D}$. We show decidability of such a weakened first--order logic, using filtration. We also investigate Craig interpolation and Beth definability of these logics. J.~van Benthem baptized the weakened version of the logic of binary relations ``arrow--logic''. This is a modal logic, interpreted on a set of arrows, with modalities for composition and inverse of arrows, and a (constant) modality denoting that the source and target of an arrow are the same. In this work, we identify arrows with the pair $<${\em source, target}$>$. We investigate the complete spectrum of arrow--logics in which the domain of the models is a binary relation, satisfying a combination of the conditions $\{$ reflexivity, symmetry, transitivity, Cartesian product $\}$. We systematically treat decidability, finite axiomatizability, Craig interpolation and Beth definability. Our results can be summarized in one sentence: an arrow--logic has one or more of these positive properties if and only if the domains of the models are not necessarily transitive relations.