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text: 'Let $\\langle P, \\leq_P \\rangle$ and $\\langle Q, \\leq_Q \\rangle$ be [3rb posets]. Then a function $\\phi : P \\rightarrow Q$ is said to be **monotone** (alternatively, **order-preserving**) if for all $s, t \\in P$, $s \\le_P t$ implies $\\phi(s) \\le_Q \\phi(t)$.\n\nPositive example\n------\n\n\n\n%%comment:\ndot source:\n\ndigraph G {\n node [width = 0.1, height = 0.1]\n rankdir = BT;\n rank = same;\n compound = true;\n fontname="MathJax_Main";\n\n subgraph cluster_P {\n node [style=filled,color=white];\n edge [arrowhead = "none"];\n style = filled;\n color = lightgrey;\n fontcolor = black;\n label = "P";\n labelloc = b;\n b -> a;\n c -> a;\n \n }\n subgraph cluster_Q {\n node [style=filled];\n edge [arrowhead = "none"];\n color = black;\n fontcolor = black;\n label= "Q";\n labelloc = b;\n u -> t;\n }\n edge [color = blue, style = dashed]\n fontcolor = blue;\n label = "φ"; \n labelloc = t; \n b -> t [constraint = false];\n a -> t [constraint = false];\n c -> u [constraint = false];\n}\n\n%%\n\nHere is an example of a monotone map $\\phi$ from a poset $P$ to another poset $Q$. Since $\\le_P$ has two comparable pairs of elements, $(c,a)$ and $(b,a)$, there are two constraints that $\\phi$ must satisfy to be considered monotone. Since $c \\leq_P a$, we need $\\phi(c) = u \\leq_Q t = \\phi(a)$. This is, in fact, the case. Also, since $b \\leq_P a$, we need $\\phi(b) = t \\leq_Q t = \\phi(a)$. This is also true.\n\nNegative example\n---------------\n\n\n\n%%comment:\ndot source:\n\ndigraph G {\n node [width = 0.1, height = 0.1]\n rankdir = BT;\n rank = same;\n compound = true;\n fontname="MathJax_Main";\n\n subgraph cluster_P {\n node [style=filled,color=white];\n edge [arrowhead = "none"];\n style = filled;\n color = lightgrey;\n fontcolor = black;\n label = "P";\n labelloc = b;\n a -> b;\n }\n\n subgraph cluster_Q {\n node [style=filled];\n edge [arrowhead = "none"];\n color = black;\n fontcolor = black;\n label= "Q";\n labelloc = b;\n w -> u;\n w -> v;\n u -> t;\n v -> t;\n }\n edge [color = blue, style = dashed]\n fontcolor = blue;\n label = "φ"; \n labelloc = t;\n b -> u [constraint = false];\n a -> v [constraint = false];\n}\n%%\n\nHere is an example of another map $\\phi$ between two other posets $P$ and $Q$. This map is not monotone, because $a \\leq_P b$ while $\\phi(a) = v \\parallel_Q u = \\phi(b)$.\n\nAdditional material\n----------------------------------\n\nFor some examples of montone functions and their applications, see [5lf]. To test your knowledge of monotone functions, head on over to [6pv].\n\n\n',
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