A non-commutative Ore LCM semigroup with arithmetic thermodynamics: the Hurwitz quaternions

The Hurwitz quaternion order provides a genuinely non-commutative right Ore, right LCM monoid whose Nica-Toeplitz system has arithmetic thermodynamics, with partition function zeta(beta) zeta(beta-1) and critical exponent 2. Conditions (C1)-(C4) for a non-injective arithmetic Bost-Connes factor are formulated, the unit-group exclusion (Remark 1.5: an infinite unit group makes the partition function diverge identically, and a finitely generated monoid has constant obstruction group) narrows the search, and (C4), non-amenability of the measured groupoid, is posed; it is settled negatively in Paper XXVIII of the series.