Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Private equity and employment - the Davos/WEF/Harvard study

The World Economic Forum at Davos has published a major study showing that workplaces of firms taken over by private equity have 10% less employees 5 years after the takeover, than other similar workplaces. The rate of plant closures, opening, acquisitions and disposals is twice as high as in other firms, and the net effect is still a job loss of 3.6%-4.5% after only 2 years, compared with other firms. Firms taken over by private equity are also more likely to go bankrupt than publicly quoted firms.