FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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
```$T_{jm}$ ($m \ge 1$ , $0 \le j \le m$ ). 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 $T_{0m}$
($m \ge 1$ ) are calculated. In particular, we obtain that the indices
of nilpotence of $T_{jm}$ are not bounded.

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Last modified: April 27, 1999