Congruences on bicyclic extensions of a linearly ordered group

  • Oleg Gutik Ivan Franko Lviv National University
  • Dušan Pagon University of Maribor
  • Kateryna Pavlyk University of Tartu
Keywords: semigroup, inverse semigroup, bicyclic semigroup, semigroup extension, linearly ordered group, group congrunece, Hölder's theorem, Hahn's theorem


In the paper we study inverse semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G) which are generated by partial monotone injective translations of a positive cone of a linearly ordered group G. We describe Green’s relations on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G), their bands and show that they are simple, and moreover, the semigroups B(G) and B^+(G) are bisimple. We show that for a commutative linearly ordered group G all non-trivial congruences on the semigroup B(G) (and B^+(G)) are group congruences if and only if the group G is archimedean. Also we describe the structure of group congruences on the semigroups B(G), B^+(G), \overline{B}(G) and \overline{B}^+(G).


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