@article{FernandesKoppitzMusunthia2019,
  author    = {Fernandes, Vitor H. and Koppitz, J{\"o}rg and Musunthia, Tiwadee},
  title     = {The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence},
  series = {Bulletin of the Malaysian Mathematical Sciences Society volume},
  volume    = {42},
  journal   = {Bulletin of the Malaysian Mathematical Sciences Society volume},
  number    = {5},
  publisher = {Malaysian mathematical sciences sciences soc},
  address   = {Pulau Punang},
  issn      = {0126-6705},
  doi       = {10.1007/s40840-017-0598-1},
  pages     = {2191 -- 2211},
  year      = {2019},
  abstract  = {A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TFn of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TFn and provide a minimal generating set for TFn. Moreover, a formula for the number of idempotents in TFn is given.},
  language  = {en}
}
@article{DimitrovaFernandesKoppitz2012,
  author    = {Dimitrova, Ilinka and Fernandes, Vitor H. and Koppitz, J{\"o}rg},
  title     = {The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain},
  series = {Publicationes mathematicae},
  volume    = {81},
  journal   = {Publicationes mathematicae},
  number    = {1-2},
  publisher = {Institutum Mathematicum Universitatis Debreceniensis, Debreceni Tudom{\´a}nyegyetem Matematikai Int{\´e}zete},
  address   = {Debrecen},
  issn      = {0033-3883},
  doi       = {10.5486/PMD.2012.4897},
  pages     = {11 -- 29},
  year      = {2012},
  abstract  = {The study of the semigroups OPn, of all orientation-preserving transformations on an n-element chain, and ORn, of all orientation-preserving or orientation-reversing transformations on an n-element chain, has began in [17] and [5]. In order to bring more insight into the subsemigroup structure of OPn and ORn, we characterize their maximal subsemigroups.},
  language  = {en}
}
@article{DimitrovaFernandesKoppitz2017,
  author    = {Dimitrova, Ilinka and Fernandes, Vitor H. and Koppitz, J{\"o}rg},
  title     = {A note on generators of the endomorphism semigroup of an infinite countable chain},
  series = {Journal of Algebra and its Applications},
  volume    = {16},
  journal   = {Journal of Algebra and its Applications},
  number    = {2},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4988},
  doi       = {10.1142/S0219498817500311},
  pages     = {9},
  year      = {2017},
  abstract  = {In this note, we consider the semigroup O(X) of all order endomorphisms of an infinite chain X and the subset J of O(X) of all transformations alpha such that vertical bar Im(alpha)vertical bar = vertical bar X vertical bar. For an infinite countable chain X, we give a necessary and sufficient condition on X for O(X) = < J > to hold. We also present a sufficient condition on X for O(X) = < J > to hold, for an arbitrary infinite chain X.},
  language  = {en}
}