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ABSTRACT
Let 𝒞ℛ denotes the variety of completely regular semigroups considered with the unary operation of inversion. The global study of the lattice of subvarieties of 𝒞ℛ depends heavily on various decompositions. Some of the most fruitful among these are induced by the kernel and the trace relations. In their turn, these relations are induced by the kernel and the trace relations on the lattice of congruences on regular semigroups. These latter admit the concepts of kernel and trace of a congruence. The kernel and the trace relations for congruences were transferred to kernel and trace relations on varieties but the kernel and trace got no analogue for varieties.
We supply here the kernel and the trace of a variety which induce the relations of their namesake. For the local and core relations, we also define the local and core of a variety. All the new concepts are certain subclasses of 𝒞ℛ. In this way, we achieve considerable similarity of the new concepts with those for congruences. We also correct errors in two published papers.
Acknowledgment
The unified formulation in Theorem 10.1 was kindly suggested by the referee.