FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 1, PAGES 283-305

**The partially ordered monoid of semigroup varieties under wreath
product**

A. V. Tishchenko

Abstract

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The decomposition of the ordered monoid of semigroup
varieties under wreath product into a five-element semilattice of
its subsemigroups is obtained.
One of these subsemigroups is the one-element and consists of
the only variety of all trivial semigroups.
The second one is an ideal with the zero multiplication
consisting of all overcommutative varieties.
The third one is the free semigroup of continuum rank
consisting of all non-trivial periodic group varieties.
The fourth one is the countable semilattice of finite
nilpotent subsemigroups $T$_{jm} ($m$³
1, $0$£ j £
m).
The fifth one is a semigroup without idempotents containing
a subsemigroup isomorphic to a free semigroup of continuum
rank.
This semigroup satisfies neither right nor left cancellation law.

It is proved that $T$_{jm} are lattice
intervals of the lattice of all semigroup varieties.
The greatest variety in the semigroup $T$_{jm} is the
non-zero idempotent of monoid of all semigroup varieties.
The description of all idempotents of this monoid is known.
The equational description for the least variety
in $T$_{jm} is found.
In conclusion, the indices of nilpotence of
semigroups $T$_{0m} ($m$³
1) are calculated.
In particular, we obtain that the indices of nilpotence of
$T$_{jm}
are not bounded.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/99/991/99116h.htm

Last modified: April 27, 1999