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The monoid of order isomorphisms between principal filters of \({\varvec{\mathbb {N}}}^n\)

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Abstract

Let n be any positive integer and be the semigroup of all order isomorphisms between principal filters of the nth power of the set of positive integers \({\mathbb {N}}\) with the product order. We study algebraic properties of the semigroup . In particular, we show that is a bisimple, E-unitary, F-inverse semigroup, describe Green’s relations on and its maximal subgroups. We show that the semigroup is isomorphic to the semidirect product of the direct nth power of the bicyclic monoid by the group of permutation . Also we prove that every non-identity congruence \({\mathfrak {C}}\) on the semigroup is a group and describe the least group congruence on . We show that every Hausdorff shift-continuous topology on is discrete and discuss embedding of the semigroup into compact-like topological semigroups.

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The authors are grateful to the referee and the editor for several useful comments and suggestions.

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  1. Faculty of Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine

    Oleg Gutik & Taras Mokrytskyi

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  1. Oleg Gutik
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Gutik, O., Mokrytskyi, T. The monoid of order isomorphisms between principal filters of \({\varvec{\mathbb {N}}}^n\). European Journal of Mathematics 6, 14–36 (2020). https://doi.org/10.1007/s40879-019-00328-5

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