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ABSTRACT. Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation kerp, is the union of the idempotent p-classes. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp,trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((pV £■)/£, kerp, (pV %)/%) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple.The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found

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A completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice £(CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on £(CR) yields a subdirect decomposition of £(CR). Using these results we show that £(CR) is arguesian. This confirms the (tacit) conjecture that £(CR) is modular.

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