@article{Sripon2023, 
author = {Kitsanachai Sripon and Ekkachai Laysirikul and Yanisa Chaiya},
title = {Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {8},
pages = {18223-18233},
keywords = {Green's relations, transformation semigroup, abundant semigroup},
url = {https://www.sciopen.com/article/10.3934/math.2023926},
doi = {10.3934/math.2023926},
abstract = {Let    T  (  X  ) be the full transformation semigroup on a nonempty set    X. For an equivalence relation    E on    X and a nonempty subset    Y of    X, let               S      ¯        E    (  X  ,  Y  )  =  {  α  ∈  T  (  X  )  :  ∀  x  ,  y  ∈  Y  ,  (  x  ,  y  )  ∈  E  ⇒  (  x  α  ,  y  α  )  ∈  E  ,  x  α  ,  y  α  ∈  Y  }  .Then              S      ¯        E    (  X  ,  Y  ) is a subsemigroup of    T  (  X  ) consisting of all full transformations that leave    Y and the equivalence relation    E on    Y invariant. In this paper, we show that              S      ¯        E    (  X  ,  Y  ) is not regular in general and determine all its regular elements. Then we characterize relations        L  ,              L        ∗  ,        R   and              R        ∗   on              S      ¯        E    (  X  ,  Y  ) and apply these characterizations to obtain the abundance on such semigroup.}
}