EXTENDING THE EHRESMANN-SCHEIN-NAMBOORIPAD THEOREM
Christopher Hollings · 2009
Abstract. We extend the ‘∨-premorphisms ’ part of the Ehresmann-Schein-Nambooripad Theorem to the case of weakly E-ample semigroups and inductive categories, following on from a result of Lawson (1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms ’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for ∧-premorphisms in the case of weakly E-ample semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case, and then draw connections between this and some results on the Szendrei expansions of both inverse semigroups and inductive groupoids. 1. Background and introduction In their study of E-unitary covers for inverse semigroups, McAlister and Reilly [27] made the following definitions: Definition 1.1. [27, Definition 3.4] Let S and T be inverse semigroups. A (∨, i)premorphism is a function θ: S → T such that (∨1) (st)θ ≤ (sθ)(tθ). Definition 1.2. [27, Definition 4.1] Let S and T be inverse semigroups. A (∧, i)premorphism is a function θ: S → T such that 1 (∧1) (sθ)(tθ) ≤ (st)θ; (∧2) ′ (sθ) −1 = s −1 θ. (We note that in [27], a (∨, i)-premorphism was termed a v-prehomomorphism, whilst in [22, p. 80], it is termed simply a prehomomorphism. In [27], a (∧, i)premorphism was called a ∧-prehomomorphism; in [22, p. 80], it is called a dual prehomomorphism.)