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"content": "<p>A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra \\(A\\) into its "basic parts". Formally, we say that \\(A\\) is a subdirect product of \\(A_1\\), \\(A_2\\), ..., \\(A_n\\) when \\(A\\) is a subalgebra of the product<br />\\[<br /> A_1\\times A_2\\times\\cdots\\times A_n<br />\\]<br />and for each index \\(1\\le i\\le n\\) we have for the projection \\(\\pi_i\\) that \\(\\pi_i(A)=A_i\\). In other words, a subdirect product "uses each component completely", but may be smaller than the full product.</p><p>A trivial circumstance is that \\(\\pi_i:A\\to A_i\\) is an isomorphism for some \\(i\\). The remaining components would then be superfluous. If an algebra \\(A\\) has the property than any way of representing it as a subdirect product is trivial in this sense, we say that \\(A\\) is "subdirectly irreducible".</p><p>Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic \\(p\\)-groups \\(\\mathbb{Z}_{p^n}\\) and the Prüfer groups \\(\\mathbb{Z}_{p^\\infty}\\).</p><p>In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (<a href=\"https://math.chapman.edu/~jipsen/posets/si_lattices92.html\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">math.chapman.edu/~jipsen/poset</span><span class=\"invisible\">s/si_lattices92.html</span></a>) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?</p><p>We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", <a href=\"https://arxiv.org/pdf/2104.06539\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/pdf/2104.06539</span><span class=\"invisible\"></span></a>), so there must be oodles of finite simple lattices out there.</p><p><a href=\"https://mathstodon.xyz/tags/UniversalAlgebra\" class=\"mention hashtag\" rel=\"tag\">#<span>UniversalAlgebra</span></a> <a href=\"https://mathstodon.xyz/tags/combinatorics\" class=\"mention hashtag\" rel=\"tag\">#<span>combinatorics</span></a> <a href=\"https://mathstodon.xyz/tags/logic\" class=\"mention hashtag\" rel=\"tag\">#<span>logic</span></a> <a href=\"https://mathstodon.xyz/tags/math\" class=\"mention hashtag\" rel=\"tag\">#<span>math</span></a> <a href=\"https://mathstodon.xyz/tags/algebra\" class=\"mention hashtag\" rel=\"tag\">#<span>algebra</span></a> <a href=\"https://mathstodon.xyz/tags/AbstractAlgebra\" class=\"mention hashtag\" rel=\"tag\">#<span>AbstractAlgebra</span></a></p>",
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"en": "<p>A fundamental result in universal algebra is the Subdirect Representation Theorem, which tells us how to decompose an algebra \\(A\\) into its "basic parts". Formally, we say that \\(A\\) is a subdirect product of \\(A_1\\), \\(A_2\\), ..., \\(A_n\\) when \\(A\\) is a subalgebra of the product<br />\\[<br /> A_1\\times A_2\\times\\cdots\\times A_n<br />\\]<br />and for each index \\(1\\le i\\le n\\) we have for the projection \\(\\pi_i\\) that \\(\\pi_i(A)=A_i\\). In other words, a subdirect product "uses each component completely", but may be smaller than the full product.</p><p>A trivial circumstance is that \\(\\pi_i:A\\to A_i\\) is an isomorphism for some \\(i\\). The remaining components would then be superfluous. If an algebra \\(A\\) has the property than any way of representing it as a subdirect product is trivial in this sense, we say that \\(A\\) is "subdirectly irreducible".</p><p>Subdirectly irreducible algebras generalize simple algebras. Subdirectly irreducible groups include all simple groups, as well as the cyclic \\(p\\)-groups \\(\\mathbb{Z}_{p^n}\\) and the Prüfer groups \\(\\mathbb{Z}_{p^\\infty}\\).</p><p>In the case of lattices, there is no known classification of the finite subdirectly irreducible (or simple) lattices. This page (<a href=\"https://math.chapman.edu/~jipsen/posets/si_lattices92.html\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">math.chapman.edu/~jipsen/poset</span><span class=\"invisible\">s/si_lattices92.html</span></a>) by Peter Jipsen has diagrams showing the 92 different nontrivial subdirectly irreducible lattices of order at most 8. See any patterns?</p><p>We know that every finite subdirectly irreducible lattice can be extended to a simple lattice by adding at most two new elements (Lemma 2.3 from Grätzer's "The Congruences of a Finite Lattice", <a href=\"https://arxiv.org/pdf/2104.06539\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"\">arxiv.org/pdf/2104.06539</span><span class=\"invisible\"></span></a>), so there must be oodles of finite simple lattices out there.</p><p><a href=\"https://mathstodon.xyz/tags/UniversalAlgebra\" class=\"mention hashtag\" rel=\"tag\">#<span>UniversalAlgebra</span></a> <a href=\"https://mathstodon.xyz/tags/combinatorics\" class=\"mention hashtag\" rel=\"tag\">#<span>combinatorics</span></a> <a href=\"https://mathstodon.xyz/tags/logic\" class=\"mention hashtag\" rel=\"tag\">#<span>logic</span></a> <a href=\"https://mathstodon.xyz/tags/math\" class=\"mention hashtag\" rel=\"tag\">#<span>math</span></a> <a href=\"https://mathstodon.xyz/tags/algebra\" class=\"mention hashtag\" rel=\"tag\">#<span>algebra</span></a> <a href=\"https://mathstodon.xyz/tags/AbstractAlgebra\" class=\"mention hashtag\" rel=\"tag\">#<span>AbstractAlgebra</span></a></p>"
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