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  "body": "\n## Definition\n\nLet $(S, \\cdot)$ be a [[semigroup|0V3CFG1KK9]] and let $\\le$ be a [[partial order|011C29Q1S1]] on $S$.\nThe triple $(S, \\cdot, \\le)$ is an *ordered semigroup* if for all $a, b, c \\in S$,\n$$a \\le b \\implies ca \\le cb \\text{ and } ac \\le bc.$$\n\n## Remarks\n\nExample: $(\\N, +)$ with the usual order is an ordered semigroup, since $a \\le b$ gives $c + a \\le c + b$ and $a + c \\le b + c$.\nNon-example: $(\\Z, \\cdot)$ with the usual order is not an ordered semigroup.\nIndeed $1 \\le 2$, but $(-1) \\cdot 1 = -1 \u003e -2 = (-1) \\cdot 2$.\nAn ordered semigroup with an identity element is an *ordered monoid*.\nSome authors require the order to be total and call the general case a partially ordered semigroup.\n",
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