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  "body": "\n## Summary\n\nThe lattice of equational theories of a signature is the set of all equational theories of that signature, ordered by inclusion.\nAn equational theory is a set of identities closed under logical consequence.\nThis lattice is complete, and it is dually isomorphic to the lattice of all varieties of the signature, because larger theories have fewer models.\nIt can be described as the lattice of fully invariant congruences of a term algebra on countably many variables.\n\n## Explanations\n\n### L1 human\n\nCollect all the identities that follow from some axioms; this is an equational theory.\nAdding axioms gives a bigger theory and a smaller class of algebras that satisfy it.\nThe theories, ordered by inclusion, form a lattice that is the upside-down picture of the lattice of varieties.\nFor example, the theory of abelian groups contains the theory of groups, since it has the extra identity $xy \\approx yx$.\n",
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