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  "body": "\n## Summary\n\nA Taylor term is an idempotent term with one identity in two variables $x, y$ for each argument place.\nThe identity for a place has $x$ in that place on one side and $y$ in that place on the other side.\nHaving a Taylor term is one of the weakest nontrivial [[Mal'cev conditions|6R0CA4JWWY]].\nFor an idempotent variety, having a Taylor term means that no nontrivial algebra in the variety has only projections as term operations.\nIt is central in the [[algebraic approach to constraint satisfaction|3EH4F5H7FT]].\nThere, a finite core structure has a polynomial-time problem when its idempotent polymorphisms include a Taylor operation.\nOtherwise its problem is NP-complete.\n\n## Explanations\n\n### L1 human\n\nA projection, such as $(x, y) \\mapsto x$, simply copies one of its inputs and carries no real information.\nA Taylor term is an operation that is provably not a projection: for every input place, some equation forces the output not to depend on that place alone.\nFor example, a binary operation $x \\cdot y$ with $x \\cdot x = x$ and $x \\cdot y = y \\cdot x$ is a Taylor term.\nCommutativity rules out both projections, because $(x, y) \\mapsto x$ is not commutative.\nAlgebras with such an operation have enough structure for strong algorithms and structure theorems.\n",
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