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A semigroup is called eventually regular if a power of each element is regular. Regular and group-bound semigroups are each eventually regular. A tolerance relation on an eventually regular semigroup is introduced, and Lallement's result for regular semigroups is generalized to eventually regular semigroups. Weakly compatible tolerances on semigroups are studied.

In this paper we describe the strong\(\mathcal{P}\)-congruences on\(\mathcal{P}\)-regular semigroups in terms of their characteristic kernels and characteristic traces.

As a generalization of Preston’s kernel normal systems, \(\mathcal{P}\)-kernel normal systems for \(\mathcal{P}\)-inversive semigroups are introduced, and strongly regular \(\mathcal{P}\)-congruences on \(\mathcal{P}\)-inversive semigroups in terms of their \(\mathcal{P}\)-kernel normal systems are characterized. These results generalize the corresponding results for \(\mathcal{P}\)-regular semigroups and \(\mathcal{P}\)-inversive semigroups.