Broken Möbius categories of $$Q_{3}$$ -type and their split inverse semigroups

A class of Möbius monoids leads us to Möbius categories of \(Q_{3}\)-type via a particular breaking process, where \(Q_{3}\) is a quiver with three arrows (atoms). In this paper we show that a quasi-commutativity regarding composable atoms uniquely determines (via a certain local congruence) a half-factorial broken Möbius category of \(Q_{3}\)-type as a quotient category of the path category of \(Q_{3}\). Some examples shed light on the development of the topic under discussion. On the other hand, the Möbius breaking process can be extended for inverse semigroups as well. The Leech inverse semigroups of the two broken Möbius categories (the path category of \(Q_{3}\) and its quotient category) are split inverse semigroups in the sense that both are unions of two proper inverse subsemigroups, that is, both are with covering numbers 2. The connections on the two planes (broken Möbius categories and split inverse semigroups) are made on one hand by a local congruence \(\varrho ^{+}\) of the path category of \(Q_{3}\), and on the other hand by a normal inverse subsemigroup \(G^{+}\) namely a gauge inverse subsemigroup. This gauge inverse semigroup is a full inverse subsemigroup of the almost Cartesian product \(B\times _{0}B_{{\mathbb {N}}}\) of the bicyciclic semigroup B and the Brandt semigroup \(B_{{\mathbb {N}}}\). Special properties of this almost Cartesian product are examined by comparison with the well known polycyclic monoid.