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Symmetric inverse semigroup
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<h2 id="finite-symmetric-inverse-semigroups">Finite symmetric inverse semigroups</h2> <p>When <i>X</i> is a finite set {1, ..., <i>n</i>}, the inverse semigroup of one-to-one partial transformations is denoted by <i>C</i><i>n</i> and its elements are called charts or partial symmetries.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The notion of chart generalizes the notion of <a href="/facts/Permutation/JSw9ukmv">permutation</a>. A (famous) example of (sets of) charts are the hypomorphic mapping sets from the <a href="/facts/Reconstruction_conjecture/sYv3VVEI">reconstruction conjecture</a> in <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>The <a href="/facts/Cycle_notation/JSw9ukmv">cycle notation</a> of classical, group-based permutations generalizes to symmetric inverse semigroups by the addition of a notion called a <i>path</i>, which (unlike a cycle) ends when it reaches the <a href="/facts/Partial_function/0vJMw0Nm">"undefined" element</a>; the notation thus extended is called <i>path notation</i>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Symmetric_group/RUksK5l5">Symmetric group</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Lipscomb, S. (1997). <i>Symmetric Inverse Semigroups</i>. AMS Mathematical Surveys and Monographs. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0627-0.</li> <li>Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). <i>Classical Finite Transformation Semigroups: An Introduction</i>. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-84800-281-4">10.1007/978-1-84800-281-4</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-84800-281-4.</li> <li>Hollings, Christopher (2014). <i>Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups</i>. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-1493-1.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Grillet, Pierre A. (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4. <a href="978-0-8247-9662-4" target="_blank">978-0-8247-9662-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hollings 2014, p. 252 - Hollings, Christopher (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. ISBN 978-1-4704-1493-1. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Ganyushkin & Mazorchuk 2008, p. v - Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). Classical Finite Transformation Semigroups: An Introduction. Springer. doi:10.1007/978-1-84800-281-4. ISBN 978-1-84800-281-4. <a href="https://doi.org/10.1007%2F978-1-84800-281-4" target="_blank">https://doi.org/10.1007%2F978-1-84800-281-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lipscomb 1997, p. 1 - Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. American Mathematical Society. ISBN 0-8218-0627-0. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lipscomb 1997, p. xiii - Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. American Mathematical Society. ISBN 0-8218-0627-0. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lipscomb 1997, p. xiii - Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. American Mathematical Society. ISBN 0-8218-0627-0. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>