Completions of filter semigroups

Abstract

A filter semigroup is a filter space with a compatible semigroup operation. In the present paper, attempts have been made to obtain T2 and regular completions of filter-semigroups. Also, a few results on regular and Cauchy modification of filter semigroups have been established.

Cauchy action on filter spaces

“…is a Cauchy map with respect to the Cauchy product [9] on G × G. Properties of filter semigroups and their completions were investigated in [17]. A filter semigroup (G, D, •) is a Cauchy group, if (G, •) is a group and D is compatible with the group operations, that is, Kχ −1 ∈ D, whenever K and χ are in D. Let F ILSG (respectively, CHG) denote the category of all objects as filter semigroups with identity element (respectively, Cauchy groups) and morphisms as Cauchy maps which are also homomorphisms.…”

“…In this paper, this question is partially answered with an introduction of the notion of the Cauchy action in Section 3, where it is shown that an equivalence relation on the filters is a G-congruence. Since an algebraically compatible group structure on a filter space induces a Cauchy structure on the space, we need to look no further than a filter semigroup [17] having a continuous action on a filter space. In Section 5, attempts have been made to investigate such an action on a filter space and a few of its modifications, while Section 4 demonstrates the interaction between a Cauchy action and the continuous action on the corresponding G-space.…”

Quasi-Completion of Filter Spaces

“…Extension theorems for filter spaces [3], regular filter spaces [10], filter semigroups [11] and Cauchy spaces ( not necessarily T 2 ) [9] have led to some interesting reflective subcategories of the categories F IL and CHY with some special type of morphisms called s-maps. In case of T 2 filter spaces, an unique extension of a Cauchy map f : (X, C) −→ (Y, D) to the corresponding completion space was possible only when the codomain was a c-filter space.…”

On finest unitary extensions of topological monoids

“…If the underlying set X of a given Cauchy space (X, S) is endowed with an associative multiplication and F 1 F 2 ∈ S for any filters F 1 , F 2 from S, then this triplet is called a semigroup Cauchy space (see [9]). Thus, if X is commutative, then (X, Σ, m) is a semigroup Cauchy space.…”

“…Each Cauchy space has many completions which have been studied in papers of many authors (see [2][3][4][5][6][7][8][9][10][11]). The most important of them is perhaps the Wyler completion possessing the universal property over other completions.…”