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On Semitopological Bicyclic Extensions of Linearly Ordered Groups

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For a linearly ordered group G , we define a subset A ⊆ G to be a shift-set if, for any x, y, z ϵ A with y < x, we get x · y-1 ··z ϵ A. We describe the natural partial order and solutions of equations on the semigroup B(A) of shifts of positive cones of A . We study topologizations of the semigroup B(A). In particular, we show that, for an arbitrary countable linearly ordered group G and a nonempty shift-set A of G , every Baire shift-continuous T1-topology τ on B(A) is discrete. We also prove that, for any linearly nondensely ordered group G and a nonempty shift-set A of G , every shift-continuous Hausdorff topology τ on the semigroup B (A) is discrete.

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Authors and Affiliations

  1. I. Franko Lviv National University, Lviv, Ukraine

    O. V. Gutik & K. M. Maksymyk

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  1. O. V. Gutik
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  2. K. M. Maksymyk
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Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 59, No. 4, pp. 31–43, October–December, 2016.

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Gutik, O.V., Maksymyk, K.M. On Semitopological Bicyclic Extensions of Linearly Ordered Groups. J Math Sci 238, 32–45 (2019). https://doi.org/10.1007/s10958-019-04216-x

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