Quotient topologies on (Boolean) topological semigroups
by
Olena Hryniv
Ivan Franko National University of Lviv

It is well known that for a closed normal subgroup H of a topological group G the quotient group G/H endowed with the quotient topology (that is the strongest topology making the quotient homomorphism q:G→ G/H continuous) is a topological group too. In the category of topological semigroups such a result is not valid even in the simplest case of the quotient semigroup S/I of a topological semigroup S by a closed ideal I ⊂ S, see [CHK] []. Nonetheless if I is a closed ideal in a locally compact s-compact topological semigroup S, then the quotient semigroup S/I is a topological semigroup [CHK] (see also [LM]). In fact, the s-compactness of S in this result can be replaced by the s-compactness of I:

Theorem 1 For any closed s-compact ideal I in a locally compact topological (inverse) semigroup S the quotinet G/I is a topological (inverse) semigroup.

The s-compactness of I is essential in this theorem because of the following

Example 1 There is a Boolean locally compact topological semigroup S and a closed discrete ideal I ⊂ S such that the intersection I∩E with the set E of idempotents of S is compact but the quotient S/I is not a topological semigroup.

A semigroup S is Boolean if it is commutative and xxx=x for all x ∈ S. Each Boolean semigroup S is an inverse Clifford semigroup, that is the union S=∪_{e ∈ E}H(e) of subgroups H(e)={x ∈ S:xx=e} parametrized by idempotents e ∈ E={x ∈ S:xx=x}

The semigroup S from the above example is the product E×H of the convergent sequence E={0}∪{1/n:n ∈ N} endowed with the operation of minimum and the free Boolean group H over a discrete uncountable space, and I={0}×H.

References:

[VHK] J.H. Carruth, J.A. Hildebrant, R.J. Koch. The theory of topological semigroups, Marsel Dekker, 1983.

[LM] J. Lawson, B. Madison. On Congruences and Cones // Math. Zeitschrift 120, (1971), 18-24.

[Hr] O. Hryniv Quotient topologies on topological semilattices // Mathematical Studii. 23(2), (2005), 136–142.

Date received: May 7, 2008