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  "body": "\n## Statement\n\nLet $\\mathcal{K}$ be a class of algebras of one signature.\nAssume that the variety $\\mathcal{V} = \\mathbf{HSP}(\\mathcal{K})$ is a [[congruence-distributive variety|9GZ7F6ETC6]].\nThen every [[subdirectly irreducible algebra|DASF939439]] in $\\mathcal{V}$ belongs to $\\mathbf{HSP}_U(\\mathcal{K})$.\nHere $\\mathbf{H}$, $\\mathbf{S}$ and $\\mathbf{P}_U$ are the [[class operators|S7MJVSP79N]] of homomorphic images, subalgebras and [[ultraproducts|AX9Q86GB8M]].\nIn particular, if $\\mathcal{K}$ is a finite set of finite algebras, then every subdirectly irreducible algebra in $\\mathcal{V}$ belongs to $\\mathbf{HS}(\\mathcal{K})$.\n\n## Remarks\n\nA subdirectly irreducible $B \\in \\mathcal{V}$ has the form $C / \\psi$, where $C$ is a subalgebra of a product $\\prod_{i \\in I} A_i$ with $A_i \\in \\mathcal{K}$.\nFor each subset $J \\subseteq I$, the kernel of the projection of $C$ onto $\\prod_{i \\in J} A_i$ is a congruence $\\eta_J$ of $C$.\nDistributivity of the congruence lattice of $C$ and the irreducibility of $\\psi$ show that the sets $J$ with $\\eta_J \\le \\psi$ contain an ultrafilter $U$ on $I$.\nThen $\\psi$ contains the congruence induced by $U$, so $B$ is a homomorphic image of a subalgebra of the ultraproduct $\\prod_{i \\in I} A_i / U$.\nWhen $\\mathcal{K}$ is a finite set of finite algebras, every ultraproduct of members of $\\mathcal{K}$ is isomorphic to a member of $\\mathcal{K}$, which gives the second claim.\n\nExample: the variety of distributive lattices is generated by the two-element lattice $\\mathbf{2}$.\nLattices are congruence distributive, so every subdirectly irreducible distributive lattice lies in $\\mathbf{HS}(\\mathbf{2})$.\nThe only such lattice with more than one element is $\\mathbf{2}$ itself.\n\n## Explanations\n\n### L1 human\n\nJónsson's lemma controls the building blocks of a variety whose congruences are distributive.\nThe irreducible algebras of the variety cannot be new: they all come from the generating algebras by ultraproducts, subalgebras and quotients.\nFor a variety generated by one finite algebra, they are quotients of subalgebras of that algebra, so up to isomorphism there are only finitely many of them.\n\n### L2 human\n\nBy Birkhoff's subdirect representation theorem, a variety is determined by its subdirectly irreducible members, so Jónsson's lemma describes the whole variety.\nIt gives a description of the [[lattice of subvarieties|GJAYCVYZYF]] of a congruence-distributive variety: each subvariety is generated by its subdirectly irreducible members.\nFor example, for a finite lattice $L$, the subvarieties of $\\mathbf{HSP}(L)$ correspond to the down-sets of the finitely many subdirectly irreducible lattices in $\\mathbf{HS}(L)$, up to isomorphism.\nHere $M$ lies below $N$ when $M \\in \\mathbf{HS}(N)$.\n",
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