On lattices of topologies of some unary algebras
by
Anna Kartashova
Volgograd State Pedagogical University

Let <A, \Omega> be an arbitrary algebra. A topology on the set A is a topology on the algebra <A, \Omega> if each operation from \Omega is continuous with respect to this topology.

The set of all topologies on the algebra <A, \Omega> forms a complete lattice where order is induced by inclusion. This lattice is called the lattice of topologies of the algebra <A, \Omega> and is denoted by Im(A).

An algebra with one unary operation is called a unar.

It is known that there is a finite algebra with two unary operations whose congruence lattice is not isomorphic to a congruence lattice of a unar (see [1, Theorem 5.6]).

We prove the similar result for the class of lattices of topologies of algebras.

Theorem. For any positive integer n there exists an algebra <A, f, g> of order 2^m+1 with two unary operations such that the lattice Im(A) is distributive, consists of 4n+1 elements and Im(A) is non-isomorphic to any lattice of topologies of unars.

[1] Johnsson J., Seifert R.L. A survey of multi-unary algebra, Mimeographed seminar notes, U.C. Berkeley, 1967.

Date received: January 28, 2003