The greatest nonuniversal congruence on a symmetric inverse semigroup (Q1078336)

This paper proves the existence of the greatest nonuniversal congruence on the inverse symmetric semigroup \(I_ X\) on a set X. In case X is a finite nonempty set, the greatest nonuniversal congruence equals the smallest semilattice congruence on \(I_ X.\) The main result of this paper, of course, is the case where X is infinite. The author gives its explicit form: if \(\alpha \in I_ X\), \(\Delta\) \(\alpha\) denotes the range of \(\alpha\). For \(\alpha,\beta \in I_ X\), we define \(D(\alpha,\beta)=\{x\in \Delta \alpha \cap \Delta \beta |\) \(x\alpha =x\beta \}\). Then the greatest nonuniversal congruence equals \[ \{(\alpha,\beta)\in I_ X\times I_ X:\quad | (\Delta \alpha \cup \Delta \beta)\setminus D(\alpha,\beta)| <| X| \}. \]