In this paper, we investigate strongly regular congruences on $E$-inversive semigroups $S$. We describe the complete lattice homomorphism of strongly regular congruences, which is a generalization of an open problem of Pastijn and Petrich for regular semigroups. An abstract characterization of left and right traces for strongly regular congruences is given. The strongly regular (sr) congruences on $E$-inversive semigroups $S$ are described by means of certain strongly regular congruence triples $({\it\gamma},K,{\it\delta})$ consisting of certain sr-normal equivalences ${\it\gamma}$ and ${\it\delta}$ on $E(S)$ and a certain sr-normal subset $K$ of $S$. Further, we prove that each strongly regular congruence on $E$-inversive semigroups $S$ is uniquely determined by its associated strongly regular congruence triple.

In this paper we determine the smallest equivalence relation on a multialgebra for which the factor multialgebra is a universal algebra satisfying a given identity. We also establish an important property for the factor multialgebra (of a multialgebra) modulo this relation.

An ${\cal L}$-isomorphism between inverse semigroups $S$ and $T$ is an isomorphism between their lattices ${\cal L}(S)$ and ${\cal L}(T)$ of inverse subsemigroups. The author and others have shown that if $S$ is aperiodic – has no nontrivial subgroups – then any such isomorphism $\Phi$ induces a bijection $\phi$ between $S$ and $T$. We first characterize the bijections that arise in this way and go on to prove that under relatively weak ‘archimedean’ hypotheses, if $\phi$ restricts to an isomorphism on the semilattice of idempotents of $S$, then it must be an isomorphism on $S$ itself, thus generating a result of Goberstein. The hypothesis on the restriction to idempotents is satisfied in many applications. We go on to prove theorems similar to the above for the class of completely semisimple inverse semigroups.