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Transformer networks have seen great success in natural language processing and machine vision, where task objectives such as next word prediction and image classification benefit from nuanced context sensitivity across high-dimensional inputs. However, there is an ongoing debate about how and when transformers can acquire highly structured behavior and achieve systematic generalization. Here, we explore how well a causal transformer can perform a set of algorithmic tasks, including copying, sorting, and hierarchical compositions of these operations. We demonstrate strong generalization to sequences longer than those used in training by replacing the standard positional encoding typically used in transformers with labels arbitrarily paired with items in the sequence. We search for the layer and head configuration sufficient to solve these tasks, then probe for signs of systematic processing in latent representations and attention patterns. We show that two-layer transformers learn reliable solutions to multi-level problems, develop signs of task decomposition, and encode input items in a way that encourages the exploitation of shared computation across related tasks. These results provide key insights into how attention layers support structured computation both within a task and across multiple tasks.