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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature and let $X$ be a countably infinite set of variables.\nAn [[equational theory|F1M51B4MDG]] of signature $\\sigma$ is a set $T$ of [[identities|PKD440G2XW]] over $X$ that contains every identity true in all models of $T$.\nThe *lattice of equational theories* of $\\sigma$ is the set of all equational theories of signature $\\sigma$, ordered by inclusion.\nFor an equational theory $T_0$, the lattice of equational theories *extending* $T_0$ is its subset of theories that contain $T_0$.\n\n## Remarks\n\nThe meet of a family of theories is their intersection.\nThe join is the smallest theory containing their union, that is, the set of consequences of the union.\nSo the lattice is a [[complete lattice|D7A0ZGZT5S]], whose top is the theory containing every identity, including $x \\approx y$.\nIdentify a theory $T$ with the set of pairs $(s, t)$ such that $s \\approx t \\in T$.\nThis makes $T$ a [[fully invariant congruence|QKGW3Z9BHW]] of the [[term algebra|3YYKGWJJND]] over $X$, and every fully invariant congruence arises this way.\nExample: the identities true in all groups form a theory, and so do those true in all abelian groups; the first is contained in the second.\nNon-example: the set $\\set{xy \\approx yx}$ alone is not a theory, because it does not contain its consequences $zw \\approx wz$ and $x \\approx x$.\nThe map $T \\mapsto \\operatorname{Mod}(T)$ is a dual isomorphism onto the lattice of varieties of signature $\\sigma$.\n",
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