On Atoms of Some Sublattice of the Lattice of Universal Semigroup Classes
by
Olga Perepelkina
Rostov-on-Don, Russia

The class K of algebraic systems of some signature is called a universal class if K defined by the set of universal formulas [1]. It is obvious that the family of all universal classes of algebraic systems of the fixed signature form a complete lattice under inclusion.

Let L be a lattice of all universal semigroup classes. The lattice L has two atoms E and N. E is the universal semigroup class generated by a trivial semigroup and N is the universal semigroup class generated by a countable cyclic semigroup. The lattice L has two sublattices L_E={ K in L | E subset or equal K} and L_N={K in L | N subset or equal K}. All atoms of the lattice L_E and some atoms of the lattice L_N was described in the paper [2].

Let A_n=<a, b | ab=ba, ab=aⁿ> be a semigroup, where n in N. A_n is a universal semigroup class generated by the semigroup A_n.

Theorem. A_n is an atom of the lattice L_N, where n in N.

It was shown in [2] that A₂ is an atom of L_N.

[1] Mal'cev A. I. Algebraic systems. M., 1970. 392 p. (In Russian).

[2] Perepelkina O. A. On some atoms of the semilattice of universal semigroup classes// Sovremennaya aldebra. Issue 4(24). Rostov-on-Don, 1999. P. 57-64. (In Russian).

Date received: May 12, 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-26.