Distributive Elements in the Lattice of Nilpotent Semigroup Varieties
by
Boris M. Vernikov
Ural State University, Ekaterinburg, Russia

An element x of a lattice L is called distributive if

x \/ (y /\ z)=(x \/ y) /\ (x \/ z) for all y, z in L.

In [1] distributive elements in the lattice of all overcommutative semigroup varieties are completele determined. Here I study such elements in the lattice of all nilpotent varieties. Let

N_k be the variety given by the identity x₁x₂ ... x_k=0 (where k is a positive integer);

N_{k, l, m} be the variety given by the identity x₁x₂ ... x_k=0 and all identities of the kind u=0 where u runs over all words of length l depending on m letters (where k, l, m are positive integers with m < l < k);

P_{k, l} be the variety given by the identity x₁x₂ ... x_k=0 and all identities of the kind x₁x₂ ... x_l=x_\pi(1)x_\pi(2) ... x_\pi(l) where \pi runs over all permutations on the set {1, 2, ..., l} (where k, l are positive integers with 2 <= l < k).

Basing on results of [2, 3] I prove the following

Theorem. If a nilpotent semigroup variety V is a distributive element in the lattice of all nilpotent semigroup varieties then V coincides with one of the varieties N_k, N_{k, l, m}, P_{k, l} or N_{k, l, m} /\ P_{k, n}.

References:

1. B. M. Vernikov, Special elements in the lattice of overcommutative semigroup varieties, Mat. Zametki, submitted [Russian].

2. B. M. Vernikov and M. V. Volkov, Lattices of nilpotent semigroup varieties, in L. N. Shevrin (ed.), Algebraic Systems and their Varieties, Ural State University, Sverdlovsk (1988) 53-65 [Russian].

3. B. M. Vernikov and M. V. Volkov, Lattices of nilpotent semigroup varieties. II, Proc. Ural State University 10 (Matem., Mechan., no.1) (1998) 13-33 [Russian].

(This work was supported by the scientific program ``Universities of Russia - basic researches'' of the Ministry of Education of Russian Federation, project No.617.)

Date received: April 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-12.