An Alternative Proof of λ-Zappa–Szép Products of Bands and Groups

In previous work, Wazzan, together with Gilbert, established a connection between the λ-Zappa–Szép product of bands and groups and Billhardt’s λ-semidirect product of inverse semigroups using automata-theoretic methods. Motivated by the need for a structural and transparent interpretation of this correspondence, we present an alternative proof based on the Ehresmann–Schein–Nambooripad theorem. The approach reformulates the construction through the associated inductive groupoid and makes the matched-pair actions explicit, so that the interaction between the band component and the group component can be followed directly at the level of arrows, identities, and composition. Within this framework, several structural features of the resulting product are clarified, including regularity, idempotents, inverse-type behavior, orthodoxness, L-unipotence, and the connection with semidirect products. The paper also consolidates related intermediate results into a unified proof route and illustrates the construction by examples and schematic figures. This groupoid-based viewpoint provides a conceptually motivated treatment of the λ-Zappa–Szép product and highlights its role as a bridge between matched-pair decompositions, bands and groups, and inverse semigroup theory.