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  "body": "\n## Summary\n\nOrder theory is the branch of mathematics that studies orders, that is, binary relations that compare elements, such as [[partial orders|011C29Q1S1]], [[total orders|0316AYH1SC]] and [[preorders|0RXS8R6K37]].\nIts basic objects are partially ordered sets and the [[monotone functions|0WFF6AZPBP]] between them.\nIt includes the theory of [[lattices|WH4SFAACAS]], [[Galois connections|MBVJ4JQXC5]] and [[closure operators|17FVAN6HXX]].\nIts ideas appear in algebra, topology, logic and computer science, wherever objects can be compared by size, inclusion or strength.\n\n## Explanations\n\n### L1 human\n\nMany collections in mathematics come with a natural way to say that one element is \"below\" another.\nNumbers are compared by size, sets by inclusion, and whole numbers by divisibility.\nOrder theory studies what all these comparisons have in common.\nA picture of a finite order is a [[Hasse diagram|00EYM7MDKS]]: dots for the elements and upward lines for the steps between them.\nFor example, the divisors $1, 2, 3, 6$ of $6$ form a diamond: $1$ at the bottom, $2$ and $3$ in the middle, $6$ at the top.\n",
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