Kleiman used the variety 𝒢𝒫 of
all groups to define two endomorphisms φ^{𝒢𝒫} and φ_{𝒢𝒫} of the lattice ℒ(ℐ) of varieties
of inverse semigroups as follows: φ^{𝒢𝒫}(𝒱) = 𝒢𝒫∨𝒱 and φ_{𝒢𝒫}(𝒱) = 𝒢𝒫∧𝒱. This
introduced two congruences ν_{1} and ν_{2} on ℒ(ℐ) which have been very important in
recent studies of ℒ(ℐ).
This paper is devoted to studying further properties of the ν_{1} and ν_{3} = ν_{1} ∩ ν_{2}
congruence classes.
The first main result establishes that each ν_{1}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 ν_{3}class has a minimum member. On the other
hand, it is shown that not all ν_{3}classes have maximum members. However, it is
established that a large class of ν_{3}classes do have maximum members. If 𝒰 is a
variety satisfying an identity of the form x^{n+1}tt^{−1}x^{−n−1} = x^{n}tt^{−1}x^{−n} then
the ν_{3}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 ν_{3}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.
