Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operations to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries---including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment---correspond to a combination of basic operations over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operations, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.