Abstract
An existence variety (or e-variety) of regular semigroups is a class of regular semigroups which is closed under P, Se, and H. This concept was introduced by T. E. Hall and independently for orthodox semigroups by J. Kaďourek and M. B. Szendrei who called them bivarieties. In this paper we show some properties of e-varieties of E-solid regular semigroups when they are also completely semisimple, combinatorial, or cryptic. Also the existence of e-free objects in e-varieties of E-solid (or locally inverse) regular semigroups enables us to, analogously to the case for varieties of inverse semigroups, determine an order-inverting one-to-one correspondence between e-invariant congruences and e-varieties of E-solid (or locally inverse) regular semigroups.
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