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Kleiman used the variety 𝒢𝒫 of
all groups to define two endomorphisms φ𝒢𝒫 and φ𝒢𝒫 of the lattice ℒ(ℐ) of varieties
of inverse semigroups as follows: φ𝒢𝒫(𝒱) = 𝒢𝒫∨𝒱 and φ𝒢𝒫(𝒱) = 𝒢𝒫∧𝒱. This
introduced two congruences ν1 and ν2 on ℒ(ℐ) which have been very important in
recent studies of ℒ(ℐ).
This paper is devoted to studying further properties of the ν1 and ν3 = ν1 ∩ ν2
congruence classes.
The first main result establishes that each ν1-class is a complete modular
sublattice of ℒ(ℐ), although, in some cases, the class may just consist of a single
element.
It is not difficult to see that each ν3-class has a minimum member. On the other
hand, it is shown that not all ν3-classes have maximum members. However, it is
established that a large class of ν3-classes do have maximum members. If 𝒰 is a
variety satisfying an identity of the form xn+1tt−1x−n−1 = xntt−1x−n then
the ν3-class containing 𝒰 has a maximum member. The condition that a
variety satisfies this identity is equivalent to a member of conditions, one
being that every member of 𝒱 is completely semisimple and such that ℋ is a
congruence.
The nature of the maximum element in these cases is very interesting. If 𝒰
satisfies the above identity, then the fundamental inverse semigroups contained in 𝒰
constitute a variety, 𝒱 say. Letting 𝒢 = 𝒢𝒫∩𝒰, the maximum element in the ν3-class
containing 𝒰 is shown to be the Mal’cev product 𝒢∘𝒱 of the varieties 𝒢 and 𝒱. It is
shown that this is not valid in general. Other properties of the Mal’cev product are
obtained.
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