Vol. 89, No. 2, 1980

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Modular sublattices of the lattice of varieties of inverse semigroups

Norman R. Reilly

Vol. 89 (1980), No. 2, 405–417

Kleiman used the variety 𝒢𝒫 of all groups to define two endomorphisms φ𝒢𝒫 and φ𝒢𝒫 of the lattice () of varieties of inverse semigroups as follows: φ𝒢𝒫(𝒱) = 𝒢𝒫∨𝒱 and φ𝒢𝒫(𝒱) = 𝒢𝒫∧𝒱. This introduced two congruences ν1 and ν2 on () which have been very important in recent studies of ().

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 (), 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+1tt1xn1 = xntt1xn 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 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.

Mathematical Subject Classification 2000
Primary: 20M07
Secondary: 08B15
Received: 30 November 1978
Published: 1 August 1980
Norman R. Reilly