https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Formal_ethicsFormal ethics - Revision history2026-09-24T05:27:37ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Formal_ethics&diff=252385&oldid=preven>RomanSpa: /* Symbolic representation */2014-04-24T10:54:27Z<p><span class="autocomment">Symbolic representation</span></p>
<a href="https://en.formulasearchengine.com/w/index.php?title=Formal_ethics&diff=252385&oldid=16007">Show changes</a>en>RomanSpahttps://en.formulasearchengine.com/w/index.php?title=Formal_ethics&diff=16007&oldid=preven>Kwamikagami: /* Symbolic representation */caps using AWB2013-01-06T22:11:20Z<p><span class="autocomment">Symbolic representation: </span>caps using <a href="/w/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a></p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Getting paid money to work online isn</del>'<del style="font-weight: bold; text-decoration: none;">t </del>the <del style="font-weight: bold; text-decoration: none;">easiest thing </del>to <del style="font-weight: bold; text-decoration: none;">do </del>in <del style="font-weight: bold; text-decoration: none;">the world</del>, <del style="font-weight: bold; text-decoration: none;">but it is possible. If this </del>is <del style="font-weight: bold; text-decoration: none;">something you wish </del>to <del style="font-weight: bold; text-decoration: none;">work with, then the tips presented above should </del>have <del style="font-weight: bold; text-decoration: none;">helped you. Take some time, do things </del>the <del style="font-weight: bold; text-decoration: none;">right way and then you can succeed</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A [[mathematics|mathematical]] object ''X'' has ''</ins>'the <ins style="font-weight: bold; text-decoration: none;">fixed-point property''' if every suitably well-behaved [[mapping (mathematics)|mapping]] from ''X'' to itself has a [[fixed point (mathematics)|fixed point]]. The term is most commonly used </ins>to <ins style="font-weight: bold; text-decoration: none;">describe [[topological space]]s on which every [[continuous (mathematics)|continuous]] mapping has a fixed point. But another use is </ins>in <ins style="font-weight: bold; text-decoration: none;">[[order theory]]</ins>, <ins style="font-weight: bold; text-decoration: none;">where a [[partially ordered set]] ''P'' </ins>is <ins style="font-weight: bold; text-decoration: none;">said </ins>to have the <ins style="font-weight: bold; text-decoration: none;">fixed point property if every [[increasing function]] on ''P'' has a fixed point</ins>.</div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let ''A'' be an object in the [</ins>[<ins style="font-weight: bold; text-decoration: none;">concrete category]] '''C'''</ins>. <ins style="font-weight: bold; text-decoration: none;"> Then ''A'' has the ''fixed</ins>-<ins style="font-weight: bold; text-decoration: none;">point property'' if every [[morphism]</ins>] <ins style="font-weight: bold; text-decoration: none;">(i.e</ins>., <ins style="font-weight: bold; text-decoration: none;">every [</ins>[<ins style="font-weight: bold; text-decoration: none;">Function (mathematics)|function]</ins>]<ins style="font-weight: bold; text-decoration: none;">) </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">f</ins>:<ins style="font-weight: bold; text-decoration: none;">A\</ins>to <ins style="font-weight: bold; text-decoration: none;">A</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">has a fixed point</ins>. <ins style="font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Examples==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===The closed interval===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The [[closed interval]] [0,1] has the fixed point property: Let ''f'':[0,1] → [0,1] be a continuous mapping</ins>. <ins style="font-weight: bold; text-decoration: none;">If ''f''(0) = 0 or ''f''(1) = 1, then our mapping has </ins>a <ins style="font-weight: bold; text-decoration: none;">fixed point </ins>at <ins style="font-weight: bold; text-decoration: none;">0 or 1</ins>. <ins style="font-weight: bold; text-decoration: none;">If not, then ''f''(0) > 0 and ''f''(1) − 1 < 0</ins>. <ins style="font-weight: bold; text-decoration: none;">Thus the function ''g''(''x'') = ''f''(''x'') − x is a continuous real valued function which is positive at ''x'' = 0 </ins>and <ins style="font-weight: bold; text-decoration: none;">negative at ''x'' = 1</ins>. <ins style="font-weight: bold; text-decoration: none;">By the </ins>[<ins style="font-weight: bold; text-decoration: none;">[intermediate value theorem]], there is some point ''x''<sub>0</sub> with ''g''(''x''<sub>0</sub>) = 0, which is to say that ''f''(''x''<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>) − ''x''<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> = 0, and so ''x''<sub>0</sub> is a fixed point</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The [[open interval]] does ''not'' have the fixed-point property</ins>. <ins style="font-weight: bold; text-decoration: none;">The mapping ''f''(''x'') = ''x''<sup>2<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup> has no fixed point on the interval (0,1).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===The closed disc===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The closed interval is a special case of the [[closed disc]</ins>], <ins style="font-weight: bold; text-decoration: none;">which in any finite dimension has the fixed-point property by </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Brouwer fixed-point theorem]]</ins>. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Topology==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A [[Deformation retract|retract]] ''A'' </ins>of <ins style="font-weight: bold; text-decoration: none;">a space ''X'' </ins>with <ins style="font-weight: bold; text-decoration: none;">the fixed-point property also has the fixed-point property</ins>. <ins style="font-weight: bold; text-decoration: none;"> This is because if <math>r:X\</ins>to <ins style="font-weight: bold; text-decoration: none;">A</math> is a retraction and <math>f:A\to A</math> is any continuous function</ins>, <ins style="font-weight: bold; text-decoration: none;">then the composition <math>i\circ f\circ r:X\</ins>to <ins style="font-weight: bold; text-decoration: none;">X</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">(where </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">i:A\</ins>to <ins style="font-weight: bold; text-decoration: none;">X</math> is inclusion) has a fixed point</ins>. <ins style="font-weight: bold; text-decoration: none;"> That is, there </ins>is <ins style="font-weight: bold; text-decoration: none;"><math>x\</ins>in <ins style="font-weight: bold; text-decoration: none;">A<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> such that <math>f\circ r(x)=x<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins>. <ins style="font-weight: bold; text-decoration: none;"> Since <math>x\in A</math> we have that <math>r(x)=x</math> and therefore <math>f(x)=x</ins>.<ins style="font-weight: bold; text-decoration: none;"><</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A topological space has the fixed-point property if and only if its identity map is [[universal map|universal]]. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A [[product (topology)| product]] of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The FPP is a [[topological invariant]], i.e. is preserved by any [[homeomorphism]]. The FPP is also preserved by any [[retraction]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">According to [[Brouwer fixed point theorem]] every [[compact space|compact]</ins>] and [<ins style="font-weight: bold; text-decoration: none;">[convex set|convex]] subset of an [[euclidean space]] has the FPP</ins>. <ins style="font-weight: bold; text-decoration: none;">Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP</ins>. <ins style="font-weight: bold; text-decoration: none;">In 1932 [[Karol Borsuk|Borsuk]] asked whether compactness together with [[contractible space|contractibility</ins>]<ins style="font-weight: bold; text-decoration: none;">] could </ins>be <ins style="font-weight: bold; text-decoration: none;">a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by [[Shin'ichi Kinoshita|Kinoshita]] who found an example of a compact contractible space without the FPP.<ref>Kinoshita, S. On Some Contractible Continua without Fixed Point Property. ''Fund. Math</ins>.<ins style="font-weight: bold; text-decoration: none;">'' '''40''' (1953), 96–98</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references/></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{cite book | first = Norman Steenrod | last = Samuel Eilenberg | title = Foundations of Algebraic Topology | publisher = Princeton University Press | year = 1952}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{cite book | first = Bernd | last = Schröder | title = Ordered Sets | publisher = Birkhäuser Boston | year = 2002}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Fixed points (mathematics)]]</ins></div></td></tr>
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