Hausdorff properties of topological algebras

Abstract

Let P be a property of topological spaces. Let [P ] be the class of all varieties V having the property that any topological algebra in V has underlying space satisfying property P . We show that if P is preserved by finite products, and if ¬P is preserved by ultraproducts, then [P ] is a class of varieties that is definable by a Maltsev condition.The property that all T 0 topological algebras in V are j-step Hausdorff (H j ) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to T 0 ⇒ H j by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.Finally, we show that the topological implication T 0 ⇒ T 2 holds in every k-permutable, congruence modular variety.

An easy test for congruence modularity

“…The question raised by Bentz was answered by Kearnes and Luís Sequeira in [18], where it was shown that any variety realizing Σ 1 must already be congruence modular.…”

A characterization of Hausdorff separation for a special class of varieties

“…Now define U according to (7). Analogous to above, we then have that f A ( U ) ⊂ U ∪ {1 * }, and hence it remains to show that 1 * ∈ f A ( U ).…”

“…In [10], he mentioned the first result involving a separation property, by showing that in congruence permutable varieties every T 0 -topological algebra is T 2 . Note that when dealing with topological properties of algebras in varieties, it suffices to consider T 0 -topological algebras, see for example [7].…”

“…As a first step towards solving this question, the author showed in [1] the validity of T 0 =⇒ T 2 in n-permutable varieties that contain a majority term. Later, Kearnes and Sequeira [7] proved that modularity is indeed a sufficient condition for this implication to hold. Concerning the necessity of modularity they showed what they called a"partial converse", showing that if T 0 =⇒ T 2 can be characterized by a combination of n-permutability and equations on the congruence lattice at all, then the right characterization is the one suggested by Coleman.…”

“…We will show that topological algebras in these varieties do satisfy the implication T 0 =⇒ T 2 if and only if they are congruence modular and n-permutable for some n. By the results of Coleman [3] and of Kearnes and Sequeira [7], it suffices to show that a variety satisfying T 0 =⇒ T 2 is congruence modular.…”