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text: 'An **order relation** (also called an **order** or **ordering**) is a [3nt binary relation] $\\le$ on a [3jz set] $S$ that can be used to order the elements in that set.\n\nAn order relation satisfies the following properties:\n\n1. For all $a \\in S$, $a \\le a$. (the [reflexive_relation reflexive] property)\n2. For all $a, b \\in S$, if $a \\le b$ and $b \\le a$, then $a = b$. (the [antisymmetric_relation antisymmetric] property)\n3. For all $a, b, c \\in S$, if $a \\le b$ and $b \\le c$, then $a \\le c$. (the [transitive_relation transitive] property)\n\nA set that has an order relation is called a [3rb partially ordered set] (or "poset"), and $\\le$ is its *partial order*.\n\n## Totality of an order\n\nThere is also a fourth property that distinguishes between two different types of orders:\n\n4. For all $a, b \\in S$, either $a \\le b$ or $b \\le a$ or both. (the [total_relation total] property)\n\nThe total property implies the reflexive property, by setting $a = b$.\n\nIf the order relation satisfies the total property, then $S$ is called a [-540], and $\\le$ is its *total order*.\n\n## Well-ordering\n\nA fifth property that extends the idea of a "total order" is that of the [55r well-ordering]:\n\n5. For every subset $X$ of $S$, $X$ has a least element: an element $x$ such that for all $y \\in X$, we have $x \\leq y$.\n\nWell-orderings are very useful: they are the orderings we can perform [mathematical_induction induction] over. (For more on this viewpoint, see the page on [structural_induction].)\n\n# Derived relations\n\nThe order relation immediately affords several other relations.\n\n## Reverse order\n\nWe can define a *reverse order* $\\ge$ as follows: $a \\ge b$ when $b \\le a$. \n\n## Strict order \n\nFrom any poset $(S, \\le)$, we can derive a *strict order* $<$, which disallows equality. For $a, b \\in S$, $a < b$ when $a \\le b$ and $a \\neq b$. This strict order is still antisymmetric and transitive, but it is no longer reflexive.\n\nWe can then also define a reverse strict order $>$ as follows: $a > b$ when $b \\le a$ and $a \\neq b$.\n\n## Incomparability\n\nIn a poset that is not totally ordered, there exist elements $a$ and $b$ where the order relation is undefined. If neither $a \\leq b$ nor $b \\leq a$ then we say that $a$ and $b$ are *incomparable*, and write $a \\parallel b$. \n\n## Cover relation\n\nFrom any poset $(S, \\leq)$, we can derive an underlying *cover relation* $\\prec$, defined such that for $a, b \\in S$, $a \\prec b$ whenever the following two conditions are satisfied:\n\n1. $a < b$.\n2. For all $s \\in S$, $a \\leq s < b$ implies that $a = s$.\n\nSimply put, $a \\prec b$ means that $b$ is the smallest element of $S$ which is strictly greater than $a$.\n$a \\prec b$ is pronounced "$a$ is covered by $b$", or "$b$ covers $a$", and $b$ is said to be a *cover* of $a$.',
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