Related Reflections to the Axioms of Separation in Semigroups with Topologies and Some Applications

Commutative Topological Semigroups Embedded into Topological Abelian Groups

“…The functor C i is studied in [10,18,19], in the category of semitopological groups. Similar results are found in [11,20], but in the category of topological semigroups. The following proposition summarizes some properties in this regard.…”

“…Let S be a topological monoid with open shifts. Then (i) C r (S) is a monoid, ϕ (S,C i ) is a homomorphism (see [20] (Proposition 3.8)) and C r (S) = C 0 (S sr ) (see [11] (Theorem 2.8)). If S is also cancellative, C r (S) is cancellative (see [11] (Lemma 4.6)).…”

“…Then (i) C r (S) is a monoid, ϕ (S,C i ) is a homomorphism (see [20] (Proposition 3.8)) and C r (S) = C 0 (S sr ) (see [11] (Theorem 2.8)). If S is also cancellative, C r (S) is cancellative (see [11] (Lemma 4.6)). (ii) If A is an open set in S, then C r (A) = ϕ (S,C r ) (A) (see [20] (Corollary 5.7) and [21] (Lemma 4)).…”

“…(iii) C 3 (S) = S sr . Moreover, if S is T 2 , C r (S) = S sr (see [11] (Corollary 2.7) and [21] (Proposition 1)). (iv) ϕ (S,C i ) is open for each i ∈ {0, 1, 2} (see [11] (Proposition 2.1)).…”

“…In Section 4, we present sufficient conditions under which a topological semigroup have countable cellularity, the reflection on the class of regular spaces allows for us to disregard the axioms of separation to obtain topological monoids with countable cellularity. The class of topological spaces having countable cellularity is wide, in fact, it contains (among other classes) the class of the σ-compact paratopological groups (see [10] (Corollary 2.3)), the class of sequentially compact σ-compact cancellative topological monoids (see [11] (Theorem 4.8)) and the class of subsemigroups of precompact topological groups (see [12] (Corollary 3.6)). However there are still open problems that are related to this topic, for example, it is unknown if a countably compact Hausdorff topological semigroup has countable cellularity (see [9] (Problem 2.5.2)).…”