@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}
}
@article{DimitrovaKoppitz2017,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On the semigroup of all partial fence-preserving injections on a finite set},
  series = {Journal of Algebra and Its Applications},
  volume    = {16},
  journal   = {Journal of Algebra and Its Applications},
  number    = {12},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4988},
  doi       = {10.1142/S0219498817502231},
  pages     = {14},
  year      = {2017},
  abstract  = {For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α-1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green's relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n-2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.},
  language  = {en}
}