Inverse semigroups generated by nilpotent transformations

Abstract

Let $X$ be a set with infinite cardinality $m$ and let $B$ be the Baer-Levi semigroup, consisting of all one-one mappings $a:X\rightarrow X$ for which $∣X\Xα∣ = m$. Let $K_m=<B^{-1}B>$, the inverse subsemigroup of the symmetric inverse semigroup $\mathcal T(X)$ generated by all products $\beta^{−1}\gamma$, with $\beta,1\gamma\in B$. Then $K_m = <N_2>$, where $N_2$ is the subset of $\mathcal T(X)$ consisting of all nilpotent elements of index 2. Moreover, $K_m$ has 2-nilpotent-depth 3, in the sense that $N_2\cup N_2^2\subset K_m = N_2\cup N_2^2\cup N_2^3$. Let $P_m$ be the ideal $\{\alpha\in K_m: ∣dom \alpha∣<m\}$ in $K_m$ and let $L_m$ be the Rees quotient $K_m/P_m$. Then $L_m$ is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image $L_m^*$ of $L_m$ also has these properties and is congruence-free.