Inverse semigroups generated by nilpotent transformations

Abstract

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.