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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nA [[congruence|DCSXXT4SER]] $\\theta$ on $\\mathbf{A}$ is *fully invariant* when, for every endomorphism $\\varepsilon$ of $\\mathbf{A}$ and all $a, b \\in A$, $(a, b) \\in \\theta$ implies $(\\varepsilon(a), \\varepsilon(b)) \\in \\theta$.\nHere an endomorphism is a [[homomorphism|KBXSZC6BV1]] from $\\mathbf{A}$ to itself.\n\n## Remarks\n\nThe congruences $\\Delta_A$ and $\\nabla_A$ are always fully invariant, and so is the intersection of any family of fully invariant congruences.\n\nExample: on the group $\\Z^2 = \\Z \\times \\Z$, congruence modulo $2$ in both coordinates is fully invariant.\nEvery endomorphism of $\\Z^2$ is given by an integer matrix $M$, and $a - b \\in 2\\Z^2$ implies $M a - M b = M(a - b) \\in 2\\Z^2$.\n\nNon-example: on $\\Z^2$, let $(a, b) \\in \\theta$ when the first coordinates of $a$ and $b$ are equal.\nThis is a congruence, but the swap $(x, y) \\mapsto (y, x)$ is an endomorphism that sends the related pair $((0, 0), (0, 1))$ to the unrelated pair $((0, 0), (1, 0))$.\n\nOn the [[term algebra|3YYKGWJJND]] $\\mathbf{T}(X)$ over a set of variables $X$, the endomorphisms are the substitutions.\nSo the fully invariant congruences on $\\mathbf{T}(X)$ are the [[equational theories|F1M51B4MDG]] in the variables $X$.\n",
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