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<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (Semigroups) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a 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/>Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:marina.anagnostopoulou-merkouri@bristol.ac.uk" >marina.anagnostopoulou-merkouri@bristol.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://marinaanagno.github.io " >https://marinaanagno.github.io</a></span >br />Email: <span class="URL" ><a href="mailto:sam@math.rwth-aachen.de" >sam@math.rwth-aachen.de</a></span >br />Homepage: <span class="URL" ><a href="https://www.math.rwth-aachen.de/~Thomas.Breuer/ " >https://www.math.rwth-aachen.de/~Thomas.Breuer/</a></span >br />Email: <span class="URL" ><a href="mailto:stuartburrell1994@gmail.com" >stuartburrell1994@gmail.com</a></span >br />Homepage: <span class="URL" ><a href="https://stuartburrell.github.io " >https://stuartburrell.github.io</a></span >br />Email: <span class="URL" ><a href="mailto:rc234@st-andrews.ac.uk" >rc234@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://reinisc.id.lv/ " >https://reinisc.id.lv/</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:tom.contileslie@gmail.com" >tom.contileslie@gmail.com</a></span >br />Homepage: <span class="URL" ><a href="https://tomcontileslie.com/ " >https://tomcontileslie.com/</a></span >br />Email: <span class="URL" ><a href="mailto:jde1@st-andrews.ac.uk" >jde1@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://github.com/Joseph-Edwards " >https://github.com/Joseph-Edwards</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:attila@egri-nagy.hu" >attila@egri-nagy.hu</a></span br />Homepage: <span class="URL" ><a href="http://www.egri-nagy.hu " >http://www.egri-nagy.hu</a></span >br />Email: <span class="URL" ><a href="mailto:le27@st-andrews.ac.uk" >le27@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://le27.github.io/Luke-Elliott/ " >https://le27.github.io/Luke-Elliott/</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:ffloresbrito@gmail.com" >ffloresbrito@gmail.com</a></span >br />Email: <span class="URL" ><a href="mailto:trf1@st-andrews.ac.uk" >trf1@st-andrews.ac.uk</a></span >br />Email: <span class="URL" ><a href="mailto:nicholas.charles.ham@gmail.com" >nicholas.charles.ham@gmail.com</a></span >br />Homepage: <span class="URL" ><a href="https://n-ham.github.io " >https://n-ham.github.io</a></span >br />Email: <span class="URL" ><a href="mailto:robert.hancock@maths.ox.ac.uk" >robert.hancock@maths.ox.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://sites.google.com/view/robert-hancock/ " >https://sites.google.com/view/robert-hancock/</a></span >br />Email: <span class="URL" ><a href="mailto:mhorn@rptu.de" >mhorn@rptu.de</a></span >br />Homepage: <span class="URL" ><a href="https://www.quendi.de/math " >https://www.quendi.de/math</a></span >br />Address : <br />Fachbereich Mathematik, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany<br />br />Email: <span class="URL" ><a href="mailto:caj21@st-andrews.ac.uk" >caj21@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://heather.cafe/ " >https://heather.cafe/</a></span br />Address : <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />br />Email: <span class="URL" ><a href="mailto:j.jonusas@gmail.com" >j.jonusas@gmail.com</a></span br />Homepage: <span class="URL" ><a href="http://julius.jonusas.work " >http://julius.jonusas.work</a></span >br />Email: <span class="URL" ><a href="mailto:obk1@st-andrews.ac.uk" >obk1@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://olexandr-konovalov.github.io/ " >https://olexandr-konovalov.github.io/</a></span >br />Address : <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />br />Email: <span class="URL" ><a href="mailto:dmitrii.pasechnik@cs.ox.ac.uk" >dmitrii.pasechnik@cs.ox.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="http://users.ox.ac.uk/~coml0531/ " >http://users.ox.ac.uk/~coml0531/</a></span >br />Address : <br />Pembroke College, St. Aldates, Oxford OX1 1DW, England<br />br />Email: <span class="URL" ><a href="mailto:markus.pfeiffer@morphism.de" >markus.pfeiffer@morphism.de</a></span >br />Homepage: <span class="URL" ><a href="https://markusp.morphism.de/ " >https://markusp.morphism.de/</a></span >br />Email: <span class="URL" ><a href="mailto:jack.schmidt@uky.edu" >jack.schmidt@uky.edu</a></span >br />Homepage: <span class="URL" ><a href="https://www.ms.uky.edu/~jack/ " >https://www.ms.uky.edu/~jack/</a></span >br />Email: <span class="URL" ><a href="mailto:sergio.siccha@gmail.com" >sergio.siccha@gmail.com</a></span >br />Email: <span class="URL" ><a href="mailto:fls3@st-andrews.ac.uk" >fls3@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://flsmith.github.io/ " >https://flsmith.github.io/</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:nthiery@users.sf.net" >nthiery@users.sf.net</a></span >br />Homepage: <span class="URL" ><a href="https://nicolas.thiery.name/ " >https://nicolas.thiery.name/</a></span >br />Email: <span class="URL" ><a href="mailto:mt200@st-andrews.ac.uk" >mt200@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://mariatsalakou.github.io/ " >https://mariatsalakou.github.io/</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:cdwensley.maths@btinternet.com" >cdwensley.maths@btinternet.com</a></span >br />Email: <span class="URL" ><a href="mailto:mw231@st-andrews.ac.uk" >mw231@st-andrews.ac.uk</a></span >br />Address : <br />Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<br />br />Email: <span class="URL" ><a href="mailto:gap@wilf-wilson.net" >gap@wilf-wilson.net</a></span br />Homepage: <span class="URL" ><a href="https://wilf.me " >https://wilf.me</a></span >br />Email: <span class="URL" ><a href="mailto:mct25@st-andrews.ac.uk" >mct25@st-andrews.ac.uk</a></span >br />Homepage: <span class="URL" ><a href="https://mtorpey.github.io/ " >https://mtorpey.github.io/</a></span >br />Address : <br />Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<br />br />Email: <span class="URL" ><a href="mailto:f.zickgraf@dashdos.com" >f.zickgraf@dashdos.com</a></span >"X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a></p>fficient methods for finitely presented semigroups and monoids, and for semigroups and monoids consisting of transformations, partial permutations, bipartitions, partitioned binary relations, subsemigroups of regular Rees 0-matrix semigroups, and matrices of various semirings including boolean matrices, matrices over finite fields, and certain tropical matrices. Semigroups contains efficient methods for creating semigroups, monoids, and inverse semigroups and monoids, calculating their Green's structure, ideals, size, elements, group of units, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and so on. It is possible to test if a semigroup satisfies a particular property, such as if it is regular, simple, inverse, completely regular, and a large number of further properties. There are methods for finding presentations for a semigroup, the congruences of a semigroup, the maximal subsemigroups of a finite semigroup, smaller degree partial permutation representations, and the character tables of inverse semigroups. There are functions for producing pictures of the Green' s structure of a semigroup, and for drawing graphical representations of certain types of elements.</p>"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>strong class="pkg" >Semigroups</strong > is free software; you can redistribute it and/or modify it, under the terms of the GNU General Public License, version 3 of the License, or (at your option ) any later, version.</p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>strong class="pkg" >Semigroups</strong > package would like to thank:</p>dl >dt ><strong class="Mark" >strong ></dt >dd ><p>who contributed to the function <code class="func" >DotString</code > (<a href="chap16_mj.html#X7F51F3CD7E13D199" ><span class="RefLink" >16.1-1</span ></a>).</p>dd >dt ><strong class="Mark" >strong ></dt >dd ><p>for their contribution to the development of the algorithms for maximal subsemigroups and smaller degree partial permutation representations.</p>dd >dt ><strong class="Mark" >strong ></dt >dd ><p>who contributed to the part of the package relating to bipartitions. We also thank the University of Western Sydney for their support of the development of this part of the package.</p>dd >dt ><strong class="Mark" >strong ></dt >dd ><p>who contributed to the code for graph inverse semigroups; see Section <a href="chap7_mj.html#X850B10D783053100" ><span class="RefLink" >7.10</span ></a>.</p>dd >dt ><strong class="Mark" >strong ></dt >dd ><p>who contributed to the attribute <code class="func" >MunnSemigroup</code > (<a href="chap7_mj.html#X78FBE6DD7BCA30C1" ><span class="RefLink" >7.2-1</span ></a>).</p>dd >dt ><strong class="Mark" >strong ></dt >dd ><p>who contributed the function <code class="func" >CharacterTableOfInverseSemigroup</code > (<a href="chap11_mj.html#X7C83DF9A7973AF6D" ><span class="RefLink" >11.15-10</span ></a>).</p>dd >dl >gie Trust for the Universities of Scotland for funding the PhD scholarships of Julius Jonušas and Wilf A. Wilson when they worked on this project; the Engineering and Physical Sciences Research Council (EPSRC) for funding the PhD scholarships of F. Smith (EP/N509759/1) and M. Young (EP/M506631/1) when they worked on this project.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"contents" name="contents" ></a></h3>div class="ContChap" ><a href="chap1_mj.html#X7D8D6DB37A0326BE" >1 <span class="Heading" >strong class="pkg" >Semigroups</strong > packagespan ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7DFB63A97E67C0A1" >1.1 <span class="Heading" >span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X8389AD927B74BA4A" >1.2 <span class="Heading" >span ></a>span >div >div >div class="ContChap" ><a href="chap2_mj.html#X82398F3785F63754" >2 <span class="Heading" >Installing <strong class="pkg" >Semigroups</strong ></span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7DA3059C79842BF3" >2.1 <span class="Heading" >For those in a hurry</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X849F6196875A6DF5" >2.2 <span class="Heading" >Compiling the kernel module</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X857CBE5484CF703A" >2.3 <span class="Heading" >Rebuilding the documentation</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7862D3F37C5BBDEF" >2.4 <span class="Heading" >Testing your installation</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X80F85B577A3DFCF9" >2.4-1 SemigroupsTestInstall</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7C2D57708006AB63" >2.4-2 SemigroupsTestStandard</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7ED2F9C784B554D8" >2.4-3 SemigroupsTestExtreme</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X8544F4BD79F0BF3C" >2.4-4 SemigroupsTestAll</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X798CBC46800AB80F" >2.5 <span class="Heading" >More information during a computation</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X85CD4E6C82BECAF3" >2.5-1 InfoSemigroups</a></span >div ></div >div >div class="ContChap" ><a href="chap3_mj.html#X7C18DB427C9C0917" >3 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7850845886902FBF" >3.1 <span class="Heading" >The family and categories of bipartitions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X80F11BEF856E7902" >3.1-1 IsBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82F5D10C85489832" >3.1-2 IsBipartitionCollection</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X85D77073820C7E72" >3.2 <span class="Heading" >Creating bipartitions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7E052E6378A5B758" >3.2-1 Bipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X846AA7568435D2CE" >3.2-2 BipartitionByIntRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8379B0538101FBC8" >3.2-3 IdentityBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X824EDD4582AAA8C7" >3.2-4 LeftOne</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X790B71108070FAC2" >3.2-5 RightOne</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7CE00E0C79F62745" >3.2-6 StarOp</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8077265981409CCB" >3.2-7 RandomBipartition</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7C2C44D281A0D2C9" >3.3 <span class="Heading" >Changing the representation of a bipartition</span span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X855126D98583C181" >3.3-1 AsBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X85A5AD2B7F3B776F" >3.3-2 AsBlockBijection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7CE91D0C83865214" >3.3-3 AsTransformation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7C5212EF7A200E63" >3.3-4 AsPartialPerm</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7C684CD38405DBEF" >3.3-5 AsPermutation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X83F2C3C97E8FFA49" >3.4 <span class="Heading" >Operators for bipartitions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A39D36086647536" >3.4-1 PartialPermLeqBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8608D78F83D55108" >3.4-2 NaturalLeqPartialPermBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79E8FA077E24C1F4" >3.4-3 NaturalLeqBlockBijection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D9F5A248028FF52" >3.4-4 PermLeftQuoBipartition</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X87F3A304814797CE" >3.5 <span class="Heading" >Attributes for bipartitons</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X780F5E00784FE58C" >3.5-1 DegreeOfBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82074756826AD2C2" >3.5-2 RankOfBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X86F6506C780C6E08" >3.5-3 ExtRepOfObj</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7ECD393A854C073B" >3.5-4 IntRepOfBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X86A10B138230C2A4" >3.5-5 RightBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B9B364379D8F4E8" >3.5-6 LeftBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79AEDB5382FD25CF" >3.5-7 NrLeftBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X86385A3C8662E1A7" >3.5-8 NrRightBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8110B6557A98FB5C" >3.5-9 NrBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8657EE2B79E1DD02" >3.5-10 DomainOfBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84569A187A211332" >3.5-11 CodomainOfBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79C556827A578509" >3.5-12 IsTransBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7F0B8ACC7C9A937F" >3.5-13 IsDualTransBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8031B53E7D0ECCFA" >3.5-14 IsPermBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X87C771D37B1FE95C" >3.5-15 IsPartialPermBipartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X829494DF7FD6CFEC" >3.5-16 IsBlockBijection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79D54AD8833B9551" >3.5-17 IsUniformBlockBijection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B87B9B081FF88BB" >3.5-18 CanonicalBlocks</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X87684C148592F831" >3.6 <span class="Heading" >Creating blocks and their attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D77092078EC860C" >3.6-1 IsBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X81302B217DCAAE6F" >3.6-2 BLOCKS_NC</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7D2CB12279623CE2" >3.6-3 ExtRepOfObj</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X787D22AE7FA69239" >3.6-4 RankOfBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8527DC6A8771C2BE" >3.6-5 DegreeOfBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X815D99A983B2355F" >3.6-6 ProjectionFromBlocks</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7A45E0067F344683" >3.7 <span class="Heading" >Actions on blocks</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B701DA37F75E77B" >3.7-1 OnRightBlocks</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A5A4AF57BEA2313" >3.7-2 OnLeftBlocks</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X876C963F830719E2" >3.8 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X810BFF647C4E191E" >3.8-1 IsBipartitionSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X80C37124794636F3" >3.8-2 IsBlockBijectionSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X79A706A582ABE558" >3.8-3 IsPartialPermBipartitionSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7DEE07577D7379AC" >3.8-4 IsPermBipartitionGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8162E2BB7CF144F5" >3.8-5 DegreeOfBipartitionSemigroup</a></span >div ></div >div >div class="ContChap" ><a href="chap4_mj.html#X85A717D1790B7BB5" >4 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7C40DA67826FF873" >4.1 <span class="Heading" >The family and categories of PBRs</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82CCBADC80AE2D15" >4.1-1 IsPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X854A9CEA7AC14C0A" >4.1-2 IsPBRCollection</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X8758C4FB81D2C2A1" >4.2 <span class="Heading" >Creating PBRs</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82A8646F7C70CF3B" >4.2-1 PBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82FE736F7F11B157" >4.2-2 RandomPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8646781B7EAE04C0" >4.2-3 EmptyPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X80D20EA3816DC862" >4.2-4 IdentityPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X847BA0177D90E9D7" >4.2-5 UniversalPBR</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X86B714987C01895F" >4.3 <span class="Heading" >Changing the representation of a PBR</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X81CBBE6080439596" >4.3-1 AsPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8407F516825A514A" >4.3-2 AsTransformation</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X795B1C16819905E8" >4.3-3 AsPartialPerm</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X86786B297FBCD064" >4.3-4 AsPermutation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X872B5817878660E5" >4.4 <span class="Heading" >Operators for PBRs</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X78EC8E597EB99730" >4.5 <span class="Heading" >Attributes for PBRs</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7DFC277E80A50C2F" >4.5-1 StarOp</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X785B576B7823D626" >4.5-2 DegreeOfPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X78302D7E81BB1E54" >4.5-3 ExtRepOfObj</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7F4C8A2B79E6D963" >4.5-4 PBRNumber</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X82FD0AB179ED4AFD" >4.5-5 IsEmptyPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7E263B2F7B838D6E" >4.5-6 IsIdentityPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7A280FC27BAD0EF0" >4.5-7 IsUniversalPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X81EC86397E098BC8" >4.5-8 IsBipartitionPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7AF425D17BBE9023" >4.5-9 IsTransformationPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7962D03186B1AFDF" >4.5-10 IsDualTransformationPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7883CD5D824CC236" >4.5-11 IsPartialPermPBR</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X85B21BB0835FE166" >4.5-12 IsPermPBR</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7ECD4BBD7A0E834E" >4.6 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8554A3F878A4DC73" >4.6-1 IsPBRSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X80FC004C7B65B4C0" >4.6-2 DegreeOfPBRSemigroup</a></span >div ></div >div >div class="ContChap" ><a href="chap5_mj.html#X82D6B7FE7CAC0AFA" >5 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7ECF673C7BE2384D" >5.1 <span class="Heading" >Creating matrices over semirings</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8711618C7A8A1B60" >5.1-1 IsMatrixOverSemiring</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X86F696B883677D6B" >5.1-2 IsMatrixOverSemiringCollection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7C1CDA817CE076FD" >5.1-3 DimensionOfMatrixOverSemiring</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7FF0B2A783BA2D06" >5.1-4 DimensionOfMatrixOverSemiringCollection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7DCA234C86ED8BD3" >5.1-5 Matrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X85426D8885431ECE" >5.1-6 AsMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X82172D747D66C8CC" >5.1-7 RandomMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X782480C686F1A663" >5.1-8 <span class="Heading" >Matrix filters</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X86233A3E86512493" >5.1-9 <span class="Heading" >Matrix collection filters</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8289FCCC8274C89D" >5.1-10 AsList</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7D21408E845E4648" >5.1-11 ThresholdTropicalMatrix</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7874559881FE8779" >5.1-12 ThresholdNTPMatrix</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X807E402687741CDA" >5.2 <span class="Heading" >Operators for matrices over semirings</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X844A32A184E5EB75" >5.3 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X84A16D4D7D015885" >5.3-1 BooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7DA524567E0E7E16" >5.3-2 AsBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X87BDB89B7AAFE8AD" ><code >5.3-3 \in</code ></a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8629FA5F7B682078" >5.3-4 OnBlist</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X85E2FD8B82652876" >5.3-5 Successors</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7E0FD5878106AB66" >5.3-6 BooleanMatNumber</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X793A1C277C1D7D6D" >5.3-7 BlistNumber</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7EEA5011862E6298" >5.3-8 CanonicalBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X794C91597CC9F784" >5.3-9 IsRowTrimBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7D22BA78790EFBC6" >5.3-10 IsSymmetricBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7C373B7D87044050" >5.3-11 IsReflexiveBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7CDAD39B856AC3E5" >5.3-12 IsTransitiveBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8570C8A08549383D" >5.3-13 IsAntiSymmetricBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7A68D87982A07C6F" >5.3-14 IsTotalBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7D9BECEA7E9B72A7" >5.3-15 IsPartialOrderBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X82EA957982B79827" >5.3-16 IsEquivalenceBooleanMat</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7E6B588887D34A0A" >5.3-17 IsTransformationBooleanMat</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X873822B6830CE367" >5.4 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X857E626783CCF766" >5.4-1 RowSpaceBasis</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8733B04781B682E5" >5.4-2 RightInverse</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X8770A88E82AA24B7" >5.5 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7BC66ECE8378068E" >5.5-1 InverseOp</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7CA636F080777C36" >5.5-2 IsTorsion</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X84F59A2687C62763" >5.5-3 Order</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X86BFFFBC87F2AB1E" >5.6 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X82EC4F49877D6EB1" >5.6-1 InverseOp</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X83663A5387042B69" >5.6-2 RadialEigenvector</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X83FCFB368743E4BA" >5.6-3 SpectralRadius</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X869F60527C2B9328" >5.6-4 UnweightedPrecedenceDigraph</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X79B614AA803BD103" >5.7 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7DC6EB0680B3E4DD" >5.7-1 <span class="Heading" >Matrix semigroup filters</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8616225581BC7414" >5.7-2 <span class="Heading" >Matrix monoid filters</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X808A4061809A6E67" >5.7-3 IsFinite</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X80C6B26284721409" >5.7-4 IsTorsion</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X873DE466868DA849" >5.7-5 NormalizeSemigroup</a></span >div ></div >div >div class="ContChap" ><a href="chap6_mj.html#X7995B4F18672DDB0" >6 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7A19D22B7A05CC2F" >6.1 <span class="Heading" >Underlying algorithms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7A3AC74C7FF85825" >6.1-1 <span class="Heading" >span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7F69D8FC7D578A0C" >6.1-2 IsActingSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7E2DE9767D5D82F7" >6.1-3 <span class="Heading" >The Froidure-Pin Algorithm</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7FEE8CFA87E7B872" >6.1-4 CanUseFroidurePin</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X79BD00A682BDED7A" >6.2 <span class="Heading" >Semigroups represented by generators</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X79A15C7C83BBA60B" >6.2-1 InverseMonoidByGenerators</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X799EBA2F819D8867" >6.3 <span class="Heading" >Options when creating semigroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X78CF5DCC7C697BB3" >6.3-1 SEMIGROUPS.DefaultOptionsRec</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X87AA2EB6810B4631" >6.4 <span class="Heading" >Subsemigroups and supersemigroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7BE36790862AE26F" >6.4-1 ClosureSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7E5B4C5A82F9E0E0" >6.4-2 SubsemigroupByProperty</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X832AEDCC7BA9E5F5" >6.4-3 InverseSubsemigroupByProperty</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X82CCC1A781650878" >6.5 <span class="Heading" >Changing the representation of a semigroup</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X838F18E87F765697" >6.5-1 IsomorphismSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X83D03BE678C9974F" >6.5-2 IsomorphismMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X80ED104F85AE5134" >6.5-3 AsSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7B22038F832B9C0F" >6.5-4 AsMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X80B7B1C783AA1567" >6.5-5 IsomorphismPermGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X870210EA7912B52A" >6.5-6 RZMSNormalization</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X80DE617E841E5BA0" >6.5-7 RMSNormalization</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7E2ECC577A1CF7CA" >6.5-8 IsomorphismReesMatrixSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X820BB66381737F2D" >6.5-9 AntiIsomorphismDualFpSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7873016586653A44" >6.5-10 EmbeddingFpMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7C3F130B8362D55A" >6.6 <span class="Heading" >Random semigroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X789DE9AB79FCFEB5" >6.6-1 RandomSemigroup</a></span >div ></div >div >div class="ContChap" ><a href="chap7_mj.html#X7C76D1DC7DAF03D3" >7 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7E42E8337A78B076" >7.1 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X84C4C81380B0239D" >7.1-1 CatalanMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85C1D4307D0F5FF7" >7.1-2 EndomorphismsPartition</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X808A27F87E5AC598" >7.1-3 PartialTransformationMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7894EE357D103806" >7.1-4 SingularTransformationSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80E80A0A83B57483" >7.1-5 <span class="Heading" >Semigroups of order-preserving transformations</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X868955247F2AFAA5" >7.1-6 EndomorphismMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X862BA1C67AA1C77C" >7.2 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X78FBE6DD7BCA30C1" >7.2-1 MunnSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X82D9619B7845CAEB" >7.2-2 RookMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85D841AE83DF101C" >7.2-3 <span class="Heading" >Inverse monoids of order-preserving partial permutations</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X876C963F830719E2" >7.3 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E4B61FF7CCFD74A" >7.3-1 PartitionMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79D33B2E7BA3073A" >7.3-2 BrauerMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8378FC8B840B9706" >7.3-3 JonesMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8458B0F7874484CE" >7.3-4 PartialJonesMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DB8CB067CBE1254" >7.3-5 AnnularJonesMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8375152F7AB52B7B" >7.3-6 MotzkinMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X83C7587C81B985BA" >7.3-7 DualSymmetricInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8301C61384168D6F" >7.3-8 UniformBlockBijectionMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8444092A7967A029" >7.3-9 PlanarPartitionMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F208DC584C0B9D1" >7.3-10 ModularPartitionMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C82B25F8441928E" >7.3-11 ApsisMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X874C945E7C61A969" >7.4 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DBB30AA83663CE8" >7.4-1 FullPBRMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X857DBF537A9A9976" >7.5 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7D4B473A7D7735E3" >7.5-1 FullMatrixMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X785924807B60F187" >7.5-2 SpecialLinearMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X860B2A4382CA8F87" >7.5-3 IsFullMatrixMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X85BACB7F81660ECC" >7.6 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B20103D84E010EF" >7.6-1 FullBooleanMatMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7A43263981F2F2AF" >7.6-2 RegularBooleanMatMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X78DF50747A28098C" >7.6-3 ReflexiveBooleanMatMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79EF0EA68782CFCA" >7.6-4 HallMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F083600787C78FF" >7.6-5 GossipMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81BBCF2E84239521" >7.6-6 TriangularBooleanMatMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7F3D0AEE79AA8C98" >7.7 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81E937B6852A9C69" >7.7-1 FullTropicalMaxPlusMonoid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85EDC03180768931" >7.7-2 FullTropicalMinPlusMonoid</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7ED2F2577CD6B578" >7.8 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X82B07E907B3A55F0" >7.8-1 TrivialSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8411EBD97A220921" >7.8-2 MonogenicSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E4DFDE27BF8B8F7" >7.8-3 RectangularBand</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7982E0667ECEB265" >7.8-4 FreeSemilattice</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X801FC1D97D832A6F" >7.8-5 ZeroSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8672CFA47CA620B2" >7.8-6 LeftZeroSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E2B20C77D47F7FB" >7.8-7 BrandtSemigroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7BB29A6779E8066A" >7.9 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B2A65F382DB36EC" >7.9-1 FreeBand</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F5658DC7E56C4A6" >7.9-2 IsFreeBandCategory</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B1CD5FC7E034B88" >7.9-3 IsFreeBand</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DECF69087BB3B16" >7.9-4 IsFreeBandElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X842839C87DAAA43C" >7.9-5 IsFreeBandElementCollection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7AEF4CD1857E7DCC" >7.9-6 IsFreeBandSubsemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X808CAEC17BF271D1" >7.9-7 ContentOfFreeBandElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7CD9426180587CA4" >7.9-8 EqualInFreeBand</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85DC5D50875E55D6" >7.9-9 GreensDClassOfElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7AD6F77E7D95C996" >7.9-10 <span class="Heading" >Operators</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X850B10D783053100" >7.10 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7A9EEFD386D6F630" >7.10-1 GraphInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8187F0FF784A82CD" >7.10-2 Range</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DEE927C83D4DFDD" >7.10-3 IsVertex</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7BFDF88B799B05A0" >7.10-4 IsGraphInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7BE287A385A058BC" >7.10-5 GraphOfGraphInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X870128E4845D6ABD" >7.10-6 IsGraphInverseSemigroupElementCollection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7BC6D5107ED09DBA" >7.10-7 IsGraphInverseSubsemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DF1ACC27CC998EB" >7.10-8 VerticesOfGraphInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87500BC782212D4A" >7.10-9 IndexOfVertexOfGraphInverseSemigroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7E51292C8755DCF2" >7.11 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F3F9DED8003CBD0" >7.11-1 FreeInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7CE4CFD886220179" >7.11-2 IsFreeInverseSemigroupCategory</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B91643B827DA6DB" >7.11-3 IsFreeInverseSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7999FE0286283CC2" >7.11-4 IsFreeInverseSemigroupElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X813A291779726739" >7.11-5 IsFreeInverseSemigroupElementCollection</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DB7DCEC7E0FE9A3" >7.11-6 CanonicalForm</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87BB5D047EB7C2BF" >7.11-7 MinimalWord</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8073A2387A42B52D" >7.11-8 <span class="Heading" >Displaying free inverse semigroup elements </span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7A55FD9A7DF21C60" >7.11-9 <span class="Heading" >Operators for free inverse semigroup elementsspan ></a>span >div ></div >div >div class="ContChap" ><a href="chap8_mj.html#X86EE8DC987BA646E" >8 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X79546641809113CE" >8.1 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X861BA02C7902A4F4" >8.1-1 DirectProduct</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X8786EFBC78D7D6ED" >8.1-2 WreathProduct</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7F035EC07AA7CD97" >8.2 <span class="Heading" > Dual semigroups </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X79F2643C8642A3B0" >8.2-1 DualSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X83403224821CD079" >8.2-2 IsDualSemigroupRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X79BAAA397FC1FA2E" >8.2-3 IsDualSemigroupElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7CB64FA378EC715B" >8.2-4 AntiIsomorphismDualSemigroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7BEA92E67A6D349A" >8.3 <span class="Heading" >Strong semilattices of semigroupsspan ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X82C3F9C9861EEDFE" >8.3-1 StrongSemilatticeOfSemigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X798DE3E581978834" >8.3-2 SSSE</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7B7B70F37C9C3836" >8.3-3 IsSSSE</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X838F24247D4DBE18" >8.3-4 IsStrongSemilatticeOfSemigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X87100CE6836DE3DB" >8.3-5 SemilatticeOfStrongSemilatticeOfSemigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X79E6C08D87984579" >8.3-6 SemigroupsOfStrongSemilatticeOfSemigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X806655138370ECFF" >8.3-7 HomomorphismsOfStrongSemilatticeOfSemigroups</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7CC4F6FE87AFE638" >8.4 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X85C00EB085774624" >8.4-1 IsMcAlisterTripleSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7B5FF3A27BB057F2" >8.4-2 McAlisterTripleSemigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7A54FDB186CD2E94" >8.4-3 McAlisterTripleSemigroupGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X8046966B7F9A1ED5" >8.4-4 McAlisterTripleSemigroupPartialOrder</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X86C0C3EF84517DAB" >8.4-5 McAlisterTripleSemigroupSemilattice</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X86D6442E85881DEA" >8.4-6 McAlisterTripleSemigroupAction</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7B4EC9FC82249A83" >8.4-7 IsMcAlisterTripleSemigroupElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X854BFB1C7BA57985" >8.4-8 McAlisterTripleSemigroupElement</a></span >div ></div >div >div class="ContChap" ><a href="chap9_mj.html#X83629803819C4A6F" >9 <span class="Heading" >span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X82D4D9A578A56A8D" >9.1 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X78E15B0184A1DC14" >9.1-1 SemigroupIdeal</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7AF9B33881D185C6" >9.1-2 Ideals</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X85D4E72B787B1C49" >9.2 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X87BB45DB844D41BC" >9.2-1 GeneratorsOfSemigroupIdeal</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8777E71A82C2BAF9" >9.2-2 MinimalIdealGeneratingSet</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7DB8699784FA4114" >9.2-3 SupersemigroupOfIdeal</a></span >div ></div >div >div class="ContChap" ><a href="chap10_mj.html#X80C6C718801855E9" >10 <span class="Heading" >'s relations span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X788D6753849BAD7C" >10.1 <span class="Heading" >'s classes and representatives span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X87558FEF805D24E1" >10.1-1 <span class="Heading" >XClassOfYClass</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81B7AD4C7C552867" >10.1-2 <span class="Heading" >GreensXClassOfElement</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7B44317786571F8B" >10.1-3 <span class="Heading" >GreensXClassOfElementNC</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7D51218A80234DE5" >10.1-4 <span class="Heading" >GreensXClasses</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X865387A87FAAC395" >10.1-5 <span class="Heading" >XClassReps</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81E5A04F7DA3A1E1" >10.1-6 MinimalDClass</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X834172F4787A565B" >10.1-7 <span class="Heading" >MaximalXClasses</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7AA3F0A77D0043FB" >10.1-8 NrRegularDClasses</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7E45FD9F7BADDFBD" >10.1-9 <span class="Heading" >NrXClasses</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8140814084748101" >10.1-10 <span class="Heading" >PartialOrderOfXClasses</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X83B0EDA57F1D2F97" >10.1-11 LengthOfLongestDClassChain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7E872C5381D0DD8A" >10.1-12 IsGreensDGreaterThanFunc</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X819CCBD67FD27115" >10.2 <span class="Heading" >span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8566F84A7F6D4193" >10.2-1 <span class="Heading" >IteratorOfXClassReps</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X867D7B8982915960" >10.2-2 <span class="Heading" >IteratorOfXClasses</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X820EF2BA7D5D53B4" >10.3 <span class="Heading" >'s classes span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X85F30ACF86C3A733" >10.3-1 <span class="Heading" >Less than for Green's classes
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