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A bisimple inverse monoid of quadruples of non-negative integers. The Möbius function
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 111-126

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The additive monoid of non-negative integers ℕ is isomorphic to the right unit submonoid of the (bisimple) bicyclic semigroup B=ℕ×ℕ. The aim of this note is to construct a similar pair of monoids (B^†=ℕ×ℕ,B^‡=ℕ×ℕ×ℕ×ℕ). The monoid B^† give rise to a bisimple inverse monoid B^‡ of quadruples of non-negative integers like as Warne's 2-dimensional bicyclic semigroup. The links with the monoid of non-negative integers ℕ and with the bicyclic semigroup may turn out to be expedient also for the computation of the corresponding Möbius functions.
Keywords: bisimple inverse monoid, bicyclic semigroup, Möbius function 
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Schwab, Emil Daniel. A bisimple inverse monoid of quadruples of non-negative integers. The Möbius function. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 111-126. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a7/