Ultrafilters on Semitopological Semigroups

Quotient space of LMC-compactification as a space of z-filters

“…LMC(S) is the largest m-admissible subalgebra of CB(S). (ε, S LMC ) is the universal compactification of S. ( Definition 4.5.1 and Theorem 4.5.2 in [1]) Now we quote some prerequisite material from [6] for the description of (S F , ε) in terms of filters. For f ∈ F, Z(f ) = f −1 ({0}) is called zero set for all f ∈ F and we denote the collection of all zero sets by Z(F ).…”

“…Proof : See Lemma 2.6 and 2.7 in [6]. Unlike in the discrete setting, there is no simple correspondence between S F and F S. F S is equipped with a topology whose base is…”

“…It is obvious that ∼ is an equivalence relation on F S. Let [p] be the equivalence class of p ∈ FS, and let F S ∼ be the corresponding quotient space with the quotient map π : Proof : See Lemma 2.11 and Lemma 2.12 in [6]. Notice that by Lemma 2.6 , we have…”

“…Stone-Čech compactifications derived from a discrete semigroup S can be considered as the spectrum of the algebra B(S), the set of bounded complex-valued functions on S, or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too, (See [6] and [7]).…”

Function lattices and compactifications

“…Here, G d denotes the group G endowed with the discrete topology. The WAP-compactification of a discrete semigroup was studied using filters by Berglund and Hindman in [4] and a treatment of semigroup compactifications using equivalence classes of z-filters was given by Tootkaboni and Riazi in [14].…”