Geodesic
Properties of the subsemigroups of the bicyclic monoid
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 311-330.

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In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.
Classification : 20M05, 20M10
Keywords: bicyclic monoid; subsemigroup; generators; defining relations; automatic structures 
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     title = {Properties of the subsemigroups of the bicyclic monoid},
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Descalço, L.; Ruškuc, N. Properties of the subsemigroups of the bicyclic monoid. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 311-330. http://geodesic.mathdoc.fr/item/CMJ_2008__58_2_a1/