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<TitlePage>
  <Title>
    Semigroups
  </Title>
  <Subtitle>
    A package for semigroups and monoids
  </Subtitle>
  <Version>
    5.5.4
  </Version>
  <Author>
    James Mitchell<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>jdm3@st-andrews.ac.uk</Email>
<Homepage>https://jdbm.me</Homepage>

  </Author>
  <Author>
    Marina Anagnostopoulou-Merkouri<Alt Only="LaTeX"><Br/></Alt>
<Email>marina.anagnostopoulou-merkouri@bristol.ac.uk</Email>
<Homepage>https://marinaanagno.github.io</Homepage>

  </Author>
  <Author>
    Thomas Breuer<Alt Only="LaTeX"><Br/></Alt>
<Email>sam@math.rwth-aachen.de</Email>
<Homepage>https://www.math.rwth-aachen.de/~Thomas.Breuer/</Homepage>

  </Author>
  <Author>
    Stuart Burrell<Alt Only="LaTeX"><Br/></Alt>
<Email>stuartburrell1994@gmail.com</Email>
<Homepage>https://stuartburrell.github.io</Homepage>

  </Author>
  <Author>
    Reinis Cirpons<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>rc234@st-andrews.ac.uk</Email>
<Homepage>https://reinisc.id.lv/</Homepage>

  </Author>
  <Author>
    Tom Conti-Leslie<Alt Only="LaTeX"><Br/></Alt>
<Email>tom.contileslie@gmail.com</Email>
<Homepage>https://tomcontileslie.com/</Homepage>

  </Author>
  <Author>
    Joseph Edwards<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>jde1@st-andrews.ac.uk</Email>
<Homepage>https://github.com/Joseph-Edwards</Homepage>

  </Author>
  <Author>
    Attila Egri-Nagy<Alt Only="LaTeX"><Br/></Alt>
<Email>attila@egri-nagy.hu</Email>
<Homepage>http://www.egri-nagy.hu</Homepage>

  </Author>
  <Author>
    Luke Elliott<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>le27@st-andrews.ac.uk</Email>
<Homepage>https://le27.github.io/Luke-Elliott/</Homepage>

  </Author>
  <Author>
    Fernando Flores Brito<Alt Only="LaTeX"><Br/></Alt>
<Email>ffloresbrito@gmail.com</Email>

  </Author>
  <Author>
    Tillman Froehlich<Alt Only="LaTeX"><Br/></Alt>
<Email>trf1@st-andrews.ac.uk</Email>

  </Author>
  <Author>
    Nick Ham<Alt Only="LaTeX"><Br/></Alt>
<Email>nicholas.charles.ham@gmail.com</Email>
<Homepage>https://n-ham.github.io</Homepage>

  </Author>
  <Author>
    Robert Hancock<Alt Only="LaTeX"><Br/></Alt>
<Email>robert.hancock@maths.ox.ac.uk</Email>
<Homepage>https://sites.google.com/view/robert-hancock/</Homepage>

  </Author>
  <Author>
    Max Horn<Alt Only="LaTeX"><Br/></Alt>
<Address>
Fachbereich Mathematik, RPTU Kaiserslautern-Landau, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany<Br/>
</Address>
<Email>mhorn@rptu.de</Email>
<Homepage>https://www.quendi.de/math</Homepage>

  </Author>
  <Author>
    Christopher Jefferson<Alt Only="LaTeX"><Br/></Alt>
<Address>
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<Br/>
</Address>
<Email>caj21@st-andrews.ac.uk</Email>
<Homepage>https://heather.cafe/</Homepage>

  </Author>
  <Author>
    Julius Jonusas<Alt Only="LaTeX"><Br/></Alt>
<Email>j.jonusas@gmail.com</Email>
<Homepage>http://julius.jonusas.work</Homepage>

  </Author>
  <Author>
    Chinmaya Nagpal<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Olexandr Konovalov<Alt Only="LaTeX"><Br/></Alt>
<Address>
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<Br/>
</Address>
<Email>obk1@st-andrews.ac.uk</Email>
<Homepage>https://olexandr-konovalov.github.io/</Homepage>

  </Author>
  <Author>
    Artemis Konstantinidi<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Hyeokjun Kwon<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Dima V. Pasechnik<Alt Only="LaTeX"><Br/></Alt>
<Address>
Pembroke College, St. Aldates, Oxford OX1 1DW, England<Br/>
</Address>
<Email>dmitrii.pasechnik@cs.ox.ac.uk</Email>
<Homepage>http://users.ox.ac.uk/~coml0531/</Homepage>

  </Author>
  <Author>
    Markus Pfeiffer<Alt Only="LaTeX"><Br/></Alt>
<Email>markus.pfeiffer@morphism.de</Email>
<Homepage>https://markusp.morphism.de/</Homepage>

  </Author>
  <Author>
    Christopher Russell<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Jack Schmidt<Alt Only="LaTeX"><Br/></Alt>
<Email>jack.schmidt@uky.edu</Email>
<Homepage>https://www.ms.uky.edu/~jack/</Homepage>

  </Author>
  <Author>
    Sergio Siccha<Alt Only="LaTeX"><Br/></Alt>
<Email>sergio.siccha@gmail.com</Email>

  </Author>
  <Author>
    Finn Smith<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>fls3@st-andrews.ac.uk</Email>
<Homepage>https://flsmith.github.io/</Homepage>

  </Author>
  <Author>
    Ben Spiers<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Nicolas Thiéry<Alt Only="LaTeX"><Br/></Alt>
<Email>nthiery@users.sf.net</Email>
<Homepage>https://nicolas.thiery.name/</Homepage>

  </Author>
  <Author>
    Maria Tsalakou<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>mt200@st-andrews.ac.uk</Email>
<Homepage>https://mariatsalakou.github.io/</Homepage>

  </Author>
  <Author>
    Chris Wensley<Alt Only="LaTeX"><Br/></Alt>
<Email>cdwensley.maths@btinternet.com</Email>

  </Author>
  <Author>
    Murray Whyte<Alt Only="LaTeX"><Br/></Alt>
<Address>
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland<Br/>
</Address>
<Email>mw231@st-andrews.ac.uk</Email>

  </Author>
  <Author>
    Wilf A. Wilson<Alt Only="LaTeX"><Br/></Alt>
<Email>gap@wilf-wilson.net</Email>
<Homepage>https://wilf.me</Homepage>

  </Author>
  <Author>
    Tianrun Yang<Alt Only="LaTeX"><Br/></Alt>

  </Author>
  <Author>
    Michael Young<Alt Only="LaTeX"><Br/></Alt>
<Address>
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland<Br/>
</Address>
<Email>mct25@st-andrews.ac.uk</Email>
<Homepage>https://mtorpey.github.io/</Homepage>

  </Author>
  <Author>
    Fabian Zickgraf<Alt Only="LaTeX"><Br/></Alt>
<Email>f.zickgraf@dashdos.com</Email>

  </Author>
  <Date>
    29 August 2025
  </Date>
  <Abstract>
              The Semigroups package is a GAP package for semigroups, and monoids.
          There are particularly efficient 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.
  </Abstract>
  <Copyright>
    ©right; by J. D. Mitchell et al.<P/>
        &Semigroups; 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.
  </Copyright>
  <Acknowledgements>
            The authors of the &SEMIGROUPS; package would like to thank:
        <List>
          <Mark>
            Manuel Delgado
          </Mark>
          <Item>
            who contributed to the function <Ref Oper = "DotString"/>.
          </Item>

          <Mark>
            Casey Donoven and Rhiannon Dougall
          </Mark>
          <Item>
            for their contribution to the development of the algorithms for
            maximal subsemigroups and smaller degree partial permutation
            representations.
          </Item>

          <Mark>
            James East
          </Mark>
          <Item>
            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.
          </Item>

          <Mark>
            Zak Mesyan
          </Mark>
          <Item>
            who contributed to the code for graph inverse semigroups; see
            Section <Ref Sect="Graph inverse semigroups"/>.
          </Item>

          <Mark>
            Yann P&#233;resse and Yanhui Wang
          </Mark>
          <Item>
            who contributed to the attribute <Ref Attr = "MunnSemigroup"/>.
          </Item>

          <Mark>
            Jhevon Smith and Ben Steinberg
          </Mark>
          <Item>
            who contributed the function
            <Ref Attr = "CharacterTableOfInverseSemigroup"/>.
          </Item>
        </List>
        We would also like to acknowledge the support of: EPSRC grant number
        GR/S/56085/01; the Carnegie Trust for the Universities of Scotland
        for funding the PhD scholarships of Julius Jonu&#353;as 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.
  </Acknowledgements>
  </TitlePage>