Tight inverse semigroups that are not bisimple. (Q1434265)

It was proved in the reviewer's paper [\textit{G. I. Zhitomirskij}, Mat. Sb., N. Ser. 73(115), 500-512 (1967; Zbl 0183.30703), English translation: Math. USSR, Sb. 2, No. 4, 445-456 (1967)] that every bisimple inverse semigroup is tight (using the terminology of the first author), that is, every of its congruence relations is uniquely determined by each of its congruence classes. The authors construct tight inverse semigroups that are neither bisimple nor congruence-free. An example of such an inverse semigroup is the set of all homotheties of closed and semi-open intervals on the positive part of the real line with respect to the usual composition of partial one-to-one transformations.