Stratified LMN-convergence tower groups and their stratified LMN-uniform convergence tower structures

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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss

Stratified LMN -convergence tower groups and their stratified LMN -uniform convergence tower structures T.M.G. Ahsanullah a,∗ , Gunther Jäger b a Department of Mathematics, King Saud University, Riyadh, Saudi Arabia b School of Mechanical Engineering, University of Applied Sciences Stralsund, 18435 Stralsund, Germany

Received 31 March 2016; received in revised form 18 January 2017; accepted 19 January 2017

This article is dedicated to Professor Robert Lowen on his 70th birthday

Abstract If L and M are frames, and N is a quantale, then using stratification mappings between frames, we introduce a category of stratified LMN -convergence tower groups – a topological category. We then prove that every stratified LMN -limit tower group induces a stratified LMN -uniform convergence tower space. Also, we introduce a category of stratified LMN -Cauchy tower groups, and show that the category of strongly normal stratified LMN -limit tower groups, is isomorphic to the category of stratified LMN -Cauchy tower groups. We provide various examples in support of our theories so far developed herein the text. © 2017 Elsevier B.V. All rights reserved. Keywords: Quantale; Topology; Convergence space; Convergence group; Convergence tower group; Cauchy convergence group; Uniform convergence space; Category theory

1. Introduction Starting with a fixed basis lattice, a frame and later, with an enriched cl-premonoid [22], we studied various convergence structures on groups, obtaining several interesting results on the category of stratified lattice-valued convergence groups [3,4] and probabilistic convergence groups [7,32]. In doing so, we used the notion of a stratified framed-valued fuzzy convergence structure studied by G. Jäger [23,40] that arose from the notion of stratified lattice-valued filters defined by Höhle and Šostak [22]; also, using probabilistic convergence structure that arises from [27,45,49]. Very recently, G. Jäger [28,29] (see also [52]) proposed a stratification mapping between frames to help build so-called s-stratified LMN -convergence tower structures, L, M being frames, and N being a complete lattice or a quantale; this stratification mapping along with some additional conditions plays a crucial role in many instances. One of the striking aspects of this theory of s-stratified LMN -convergence tower spaces is that it captures various con* Corresponding author. Fax: +966 11 4676512.

E-mail addresses: [email protected] (T.M.G. Ahsanullah), [email protected] (G. Jäger). http://dx.doi.org/10.1016/j.fss.2017.01.011 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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vergence structures such as probabilistic convergence spaces [6,27,45,48], convergence spaces [11,43], lattice-valued convergence spaces [12,14,15,21,23,24,37,38,41,42,50,51]. The motive behind the present article is to explore the use of stratification mappings between different frames in an attempt to study various s-stratified LMN -convergence structures on groups and look at the categories of s-stratified LMN -convergence tower groups, s-stratified LMN -limit tower groups, s-stratified Cauchy tower groups in a more general fashion than the attempt made previously on various convergence structures in conjunctions with groups. This, however, paves the way to overlooking the previous work on lattice-valued convergence groups and beyond from a common view point. We organized our work as follows: in a preliminary section we put some basic facts of lattice theory including the notion of stratification mapping with examples that dominate the article; and in Sections 3 and 4, we continue to collect the notions of LM-filters along with some related results, and also, provide some essential results on s-stratified LMN -convergence structures introduced and studied in [28,29]. In Section 5, we study s-stratified LMN -Cauchy tower spaces. In Section 6, we introduce a notion of s-stratified LMN -convergence tower groups, some related properties and characterization, while Section 7 deals with s-stratified LMN -uniformizability of s-stratified LMN -limit tower groups, and in Section 8, we introduce a notion of s-stratified LMN -Cauchy tower group and study the link between this notion and the notion of s-stratified LMN -limit tower group. In each section, we provide a good number of stimulating examples in support of our endeavor. 2. Preliminaries Let L = (L, ≤, ∧) be a complete lattice,[19,33]. If,moreover, a complete  lattice N isequipped with a semigroup operation ∗ : N ×N −→ N satisfying α ∗( i∈J βi ) = i∈J (α ∗βi ) and ( i∈J βi ) ∗α = i∈J (βi ∗α), then (N, ≤, ∗) is called a quantale. A quantale (N, ≤, ∗) is called commutative if the underlying semigroup (N, ∗) is commutative; furthermore, if it satisfies N ∗ α = α ∗ N = α, ∀α ∈ N , then it is called integral [20,22,46]. Unless otherwise mentioned, we consider herein this text that (N, ≤, ∗) is a commutative integral quantale. We denote the category of commutative integral quantales and quantale-homomorphisms by Quant. Typical examples of quantales are the following. • For a left-continuous t-norm ∗ on [0, 1] (see [36]), ([0, 1], ≤, ∗) is a quantale. • Let + be the set of distance distribution functions, see [47]. If we consider a sup-continuous triangle function τ : + × + −→ + , then (+ , ≤, τ ) is a quantale. • Consider [0, ∞] with the opposite order and the usual addition, extended by x + ∞ = ∞ + x = ∞, then ([0, ∞], ≥, +) is a quantale. • A frame (L, ≤, ∧) is a quantale with ∗ = ∧. It is commutative, idempotent, and integral. Let Frm denote the category of frames and frame-homomorphisms. We note that for any quantale we have α ∗ β ≤ α ∧ β and that if the quantale operation is idempotent, i.e., α ∗ α = α for all α ∈ L, then ∗ = ∧. For a commutative quantale and, so for a frame, the implication operation → : L × L −→ L, called residuum is given  by α → β = {γ ∈ L : α ∗ γ ≤ β}. For a detailed account on this operation we refer to [20,22,36]. For a frame L with top element L and bottom element ⊥L and a set X, we denote the set of all L-sets on X by LX (= {ν : X −→ L}). If A ⊆ X, then a constant L-set with value α ∈ L, is denoted by αA , and is defined L as αA (x) = α, if x ∈ A and αA (x) = ⊥L , elsewhere. In particular, we write L X , resp. ⊥X , for the constant L-sets L L X are from L to L ,  i.e., we define (ν ∧ μ)(x) = with value  , resp. ⊥ . The lattice operations   extended  pointwise ν(x) ∧ μ(x), (ν ∨ μ)(x) = ν(x) ∨ μ(x), (x) = (x) = ν (x)), ν a mapping (ν i i i i∈J i∈J i∈J i∈J (νi (x)). For  f : X −→ Y , ν ∈ LX and μ ∈ LY , the image of ν under f , f → (ν) ∈ LY , is defined by f → (ν)(y) = f (x)=y ν(x), for y ∈ Y , and the pre-image of μ under f , f ← (μ) ∈ LX , is defined by f ← (μ) = μ ◦ f . If ν ∈ LX and s : L −→ M is a mapping between the frames L and M, we define s(ν) ∈ M X by s(ν)(x) = s(ν(x)), x ∈ X. If L, M ∈ |Frm|, then a mapping s : L −→ M is called a stratification mapping if it satisfies (SM1) s(⊥L ) = ⊥M , (SM2) s(L ) = M and (SM3) s(α ∧ β) = s(α) ∧ s(β) for all α, β ∈ L, where ⊥L is the bottom element in L, similarly ⊥M is the bottom element in M; while L denotes top element in L and M is the top element in M. Note that due to (SM3), the stratification mapping s is non-decreasing.

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Example 2.1. (1) Every frame morphism s : L −→ M is a stratification mapping. (2) The pointwisely smallest stratification mapping is given by  M , if α = L ; s0 (α) = ⊥M , if α = L . (3) If ⊥L is prime, then the pointwisely largest stratification mapping is given by  ⊥M , if α = ⊥L ; s1 (α) = M , if α = ⊥L . (4) A Galois connection between the frames L, M is a pair (t, s) of non-decreasing mappings s : L −→ M, t : M −→ L such that for all α ∈ L and all β ∈ M, β ≤ s(α) is equivalent to t (β) ≤ α, [19]. In this case s is called the upper adjoint and t is called the lower adjoint. Moreover, either of the mappings uniquely determines the other and s preserves arbitrary meets and t preserves arbitrary joins. Hence in this case s is a stratification mapping. A pair of non-decreasing mappings (t, s) is a Galois connection between L, M if and only if t ◦ s ≤ idL and idM ≤ s ◦ t . If s is injective, then t ◦ s = idL . However, note that t does not necessarily preserve finite meets, i.e. is not a stratification mapping in general. (5) Let L, M be completely distributive lattices and let L be furthermore equipped with an order-reversing involution  . For a non-zero coprime element α ∈ L, define sα : L −→ M by sα (β) = M if α  β  and sα (β) = ⊥M otherwise. Then sα is a stratification mapping, [31]. 3. s-Stratified LM-filters Definition 3.1. [28,29]. Let s : L −→ M be a stratification mapping. Then the map F : LX −→ M is called an s-stratified LM-filter on X if the conditions below are satisfied: (F1) (F2) (F3) (Fs)

L M M F(⊥L X ) = ⊥ , F(X ) =  ; X if ν1 , ν2 ∈ L and ν1 ≤ ν2 , then F(ν1 ) ≤ F(ν2 ); F(ν1 ) ∧ F(ν2 ) ≤ F(ν1 ∧ ν2 ), for all ν1 , ν2 ∈ LX ; s(α) ≤ F(αX ) or equivalently, s(α) ∧ F(ν) ≤ F(αX ∧ ν) for all α ∈ L and ν ∈ LX .

s (X). This set is endowed with the pointwise order, i.e., The set of all s-stratified LM-filters on X is denoted by FLM X It follows easily that for any family of s-stratified LM-filters, the we define  F ≤ G ⇔ F(ν) ≤ G(ν), for all ν ∈ L .    meet i∈J Fi is calculated as i∈J Fi (ν) = i∈J (Fi (ν) . It also follows easily that i∈J Fi exists if and only L if for each finite subfamily one obtains: Fi1 (ν1 ) ∧ ... ∧ Fin (νn ) = ⊥M , whenever ν1 ∧ ... ∧ νn = ⊥X . In particular, s X for F, G ∈ FLM (X), F ∨ G exits if and only if one has, for any ν ∈ L : (F ∨ G)(ν) = {F(ν1 ) ∧ G(ν2 ) : ν1 , ν2 ∈ M LX , ν1 ∧ ν2 ≤ ν}, whenever ν1 ∧ ν2 = ⊥L X implies F(ν1 ) ∧ G(ν2 ) = ⊥ . s (X), is called the s-stratified point Example 3.2. (1) We define, for ν ∈ LX , [x]s (ν) = s(ν(x)). Then [x]s ∈ FLM s LM-filter of x. Note that in case X = {x}, F ∈ FLM (X) implies [x]s (ν) = s(ν(x)) ≤ F(ν(x)) = F(ν) for all ν ∈ LX , i.e., [x]s ≤ F . (2) L = M, and s = idL . A stratified L-filter [22] is an idL -stratified LL-filter. (3) L = M = {0, 1}. A filter F ∈ F(X) can be identified with an s0 -stratified LL-filter (in this case, s0 = id{0,1} the only possible stratification mapping). (4) L = {0, 1}. An s-stratified LM-filter is an M-filter of ordinary subsets, [21]. (The property (M3) is always true for any mapping s that satisfies (M1) and (M2).) (5) L = [0, 1], M = {0, 1}. An s-stratified LM-filter can be identified with a prefilter, [9,38]. Note that only s = s0 is possible in order that all prefilters are s-stratified. As a consequence we should define the point prefilter by [x]s0 = {ν ∈ [0, 1]X : ν(x) = 1}. (6) Let L, M be completely distributive lattices and let L be furthermore equipped with an order-reversing involution  . For a non-zero coprime element α ∈ L, we define sα as in Example 2.1(5). An sα -stratified LM-filter is an α-enriched LM-filter in the terminology of B. Pang and Y. Zhao, [41] (we refer to [31] for further details).

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  s (X), then f ⇒ (F) ∈ F s (Y ) and is defined by: f ⇒ If f : X −→ Y is a mapping and F ∈ FLM (F) (μ) = LM s (Y ), then for any ν ∈ LX , F (f ← (μ))  = F(μ ◦ f ) for all μ ∈ LY ; and if G ∈ FLM ⇐ Y ← G (μ) : μ ∈ L , f (μ) ≤ ν , is an s-stratified LM-filter if and only if G(μ) = ⊥M , whenever f (G)(ν) = L ← f (μ) = ⊥X . s (X) and G ∈ F s (Y ), then their product F × G : LX×Y −→ M is defined for all μ ∈ LX×Y by If F ∈ FLM LM  (F × G)(μ) = {F (ν1 ) ∧ G(ν2 )|ν1 ∈ LX , ν2 ∈ LY , ν1 × ν2 ≤ μ}, where ν1 × ν2 (x, y) = ν1 (x) ∧ ν2 (y). Lemma 3.3. [29] Let s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . s (X) and G ∈ F s (Y ), then F × G ∈ F s If F ∈ FLM LM LM (X × Y ). We provide below some examples for pairs of stratification mappings. Example 3.4. (1) Let L, M be frames with prime bottom elements and define s : L −→ M and t : M −→ L as the pointwisely largest stratification mappings (see Example 2.1(3)). Then s ◦ t ≥ idM and t ◦ s ≥ idL . (2) Let M = + be the set of distance distribution functions, i.e. ϕ : [0, ∞] −→ [0, 1] is in + if and only if ϕ(0) = 0, ϕ(∞) = 1 and ϕ is left-continuous, i.e. for all x ∈ (0, ∞) we have ϕ(x) = sup{ϕ(y) : y < x}, [47]. With the pointwise minimum and supremum, then + is a frame with smallest element ε∞ defined by ε∞ (x) = 0 for 0 ≤ x < ∞ and ε∞ (∞) = 1 and largest element ε0 defined by ε0 (0) = 0 and ε0 (x) = 1 for 0 < x ≤ ∞. We consider further the interval L = [0, ∞] with the opposite order and define the mapping s : [0, ∞] −→ + by s(α) = εα with the distance distribution function εα (x) = 0 if 0 ≤ x ≤ α and εα (x) = 1 for α < x ≤ ∞. Then s(⊥L ) = s(∞) = ∞ = ⊥M and s(L ) = s(0) = 0 = M and s(α ∧ β) = s(max{α, β}) = εmax{α,β} = εα ∧ εβ and hence s is non-decreasing. The mapping t :  −→ [0, ∞], t (ϕ) = sup{α ∈ [0, ∞] : ϕ(α) = 0} satisfies s ◦ t (ϕ) ≥ ϕ for all ϕ ∈ + and t ◦ s(α) = t (εα ) = α. Moreover, we have for ϕ, ψ ∈ + that ϕ ∧ ψ(x) = 0 if and only if ϕ(x) = 0 or ψ(x) = 0 and hence t (ϕ ∧ ψ) = max{t (ϕ), t (ψ)} = t (ϕ) ∧ t (ψ). We thus have a pair of stratification mappings that satisfy s ◦ t ≥ idM and t ◦ s ≥ idL . s (X × X), we define −1 (λ) = We denote s-stratified LM-filters on X × X by , : LX×X −→ M. For ∈ FLM −1 −1 (λ ) with λ (x, y) = λ(y, X. Further, we define the map ◦  : LX×X −→ M for any  x) for all (x, y) ∈ X ×X×X X×X by ◦ (λ) = { (η) ∧ (γ ) : η, γ ∈ L , η ◦ γ ≤ λ}, where η ◦ γ (x, y) = z∈X η(x, z) ∧ γ (z, y), λ∈L for all (x, y) ∈ X × X. s (X × X). Then Lemma 3.5. [30] Let , ∈ FLM s (X × X); (a) −1 ∈ FLM s (X × X) if and only if λ ◦ γ = ⊥L M (b) ◦ ∈ FLM X×X implies (λ) ∧ (γ ) = ⊥ . s (X), Let now (X, ·) be a group with e as the identity. Let s : L −→ F, G ∈ FLM  M be a stratification mapping, X X M is defined by F  G(ν) = {F (ν1 ) ∧ G(ν2 )|ν1 , ν2 ∈ L , ν1  ν2 ≤ ν}, where then the map F  G : L −→  for any z ∈ X, ν1  ν2 (z) = xy=z ν1 (x) ∧ ν2 (y). Also, F −1 : LX −→ M is defined by F −1 (ν) = F(ν −1 ), where ν −1 : X −→ L, x → ν(x −1 ). If m : X ×X −→ X, (x, y) → xy, then for any ν1 , ν2 ∈ LX and z ∈ X, → m (ν1 × ν2 ) (z) = m(x,y)=z (ν1 × ν2 ) (x, y) = xy=z (ν1 (x) ∧ ν2 (y)) = ν1  ν2 (z). Hence F  G = m⇒ (F × G) and we have the following result.

Lemma 3.6. [3] Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that s (X) and m : X × X −→ X, (x, y) → xy, j : X −→ s ◦ t ≥ idM , t ◦ s ≥ idL . If moreover, (X, ·) is a group, F, G ∈ FLM X, x → x −1 are the group operations, then m⇒ (F × G) = F G and j ⇒ (F) = F −1 are s-stratified LM-filters on X. Lemma 3.7. Let L, M ∈ |Frm|. Let s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM s (X), and x, y ∈ X. and t ◦ s ≥ idL and s preserve arbitrary joins. Furthermore, let (X, ·) be a group, F, G ∈ FLM Then the following assertions are true:

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(a) F −1  F ≤ [e]s and F  F −1 ≤ [e]s ; (b) (F ∧ G)−1 = F −1 ∧ G −1 and (F −1 )−1 = F ; (c) (F  G)  H = F  (G  H); (d) (F  H) ∧ (G  H) = (F ∧ G)  H; (e) [x]s  [y]s = [xy]s ; (f ) [x]s  [x −1 ]s = [e]s , and [x −1 ]s = ([x]s )−1 ; (g) ([x]s ∧ F)  ([y]s ∧ G) = [xy]s ∧ (F  G); (h) [e]s  F = F  [e]s = F ; (i) if (Y, ·) is a group and f : (X, ·) −→ (Y, ·) a group homomorphism, then f ⇒ (F  G) = f ⇒ (F)  f ⇒ (G), for s (X), and any F, G ∈ FLM (j ) f ⇒ (F)−1 = f ⇒ (F −1 ). Proof. We only proof (e), the rest follows in a similar way. However, for (g), see f.i. Lemma 4.1, [3] coupled with Lemma 3.6. Let ν ∈ LX and x, y∈ X, then X ([x] s  [y]s )(ν) = {[x]s (ν1 ) ∧ [y]s (ν2 )|ν1 , ν2 ∈ L , ν1  ν2 ≤ ν} = {s(ν1 (x)) ∧ s(ν2 (y))|ν1 , ν2 ∈ LX , ν1  ν2 ≤ ν} = {s (ν1 (x) ∧ ν2 (y)) |ν1 , ν2 ∈ LX , ν1  ν2 ≤ ν} (by (SM3)) X ≤ {s ((ν1  ν2 )(xy)) |ν1 , ν2 ∈XL , ν1  ν2 ≤ν} =s {ν1  ν2 (xy)|ν1 , ν2 ∈ L , ν1  ν2 ≤ ν} ≤ s(ν(xy)) = [xy]s (ν). Conversely, we have [xy]s (ν) = s (ν(xy)) = M ∧ s (ν(xy)) L X = s(L {x} )(x) ∧ s (ν(xy)) (where s({x} ) ∈ M ) L = s({x} (x)) ∧ s (ν(xy))  = s L (x) ∧ ν(xy) (by (SM3))

{x}  L L = s L {x} (x) ∧ ({x −1 }  ν )(y) (since ν(xy) = ({x −1 }  ν)(y))      ≤ s {ν1 (x) ∧ ν2 (y)|ν1  ν2 ≤ ν} = ν1 ν2 ≤ν s ((ν1 (x)) ∧ s(ν2 (y))  = ν1 ν2 ≤ν [x]s (ν1 ) ∧ [y]s (ν2 ) 2 = [x]s  [y]s (ν).

4. s-Stratified LMN -convergence tower spaces and s-stratified LMN -uniform convergence tower spaces Definition 4.1. [29] Let L, M ∈ |Frm|, N be a complete lattice and s : L −→ M be a stratification mapping. Then the s (X) −→ P (X), is called an s-stratified LMN -convergence tower space if pair (X, q), where q = (qα )α∈N , qα : FLM the following conditions are fulfilled: (CT1) (CT2) (CT3) (CT4)

x ∈ qα ([x]s ) for all x ∈ X and α ∈ N ; s (X) with F ≤ G implies q (F) ≤ q (G); F, G ∈ FLM α α α ≤ β implies qβ (F) ⊆ qα (F); q⊥N (F) = X.

If, moreover, (N, ≤, ∗) ∈ |Quant| and (X, q) satisfies axiom (CT5): s (X), qα (F) ∩ qβ (G) ≤ qα∗β (F ∧ G) , ∀α, β ∈ N and ∀F, G ∈ FLM LMN -limit tower space. then the pair (X, q) is said tobe an s-stratified   A mapping f : (X, q) −→ X , q between two s-stratified LMN -convergence tower spaces (resp. s-stratified s (X). LMN -limit tower spaces) is called continuous if f (qα (F)) ⊆ qα (f (F)) for all α ∈ N and F ∈ FLM The category of all s-stratified LMN -convergence tower spaces (resp. s-stratified LMN -limit tower spaces) and con-

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tinuous mappings is denoted by sLMN -ConvTS (resp. sLMN -LimTS). An s-stratified LMN  -convergence tower space (X, q = (qα )α∈N ) is called left-continuous if (CTL) for A ⊆ N , β∈A qβ (F) ⊆ q∨A (F) is fulfilled. The category of s-stratified left-continuous LMN -convergence tower spaces and continuous mappings is denoted by sLMN -LCConvTS. Remark 4.2.  Let L, M ∈ |Frm|, and N be a complete lattice. If (X, q) ∈ |sLMN -LCConvTS| and if we define s (X) and α ∈ N , then (X, lim ) is an s-stratified limq F(x) = {α ∈ N : x ∈ qα (F)}, for all x ∈ X, F ∈ FLM q s X LMN -convergence spaces, i.e., limq : FLM (X) −→ N , that satisfies the following axioms. (CS1) lim[x]s (x) = N , for all x ∈ X; s (X) with F ≤ G implies lim F(x) ≤ lim G(x), for all x ∈ X. (CS2) F, G ∈ FLM Conversely, if (X, lim) is an s-stratified LMN -convergence space, then with the definition x ∈ qαlim (F) ⇐⇒ lim F(x) ≥ α we obtain  a left-continuous LMN -convergence tower space (X, q lim ). If we call a mapping s (X) and f : (X, lim) −→ X  , lim between s-stratified LMN -convergence spaces continuous if for all F ∈ FLM  x ∈ X, lim F(x) ≤ lim (f (F)) (x) and denote the category of s-stratified LMN -convergence spaces and continuous mappings by sLMN -ConvS, then the above constructions yield an isomorphism between the categories sLMN -LCConvTS and sLMN -ConvS. It can be seen from [29] that sLMN -ConvTS is a topological category, [1,44]. The initial structure for a source (fi : X −→ (Xi , qi ))i∈J  is given by: x ∈ qα (F) ⇔ fi (x) ∈ qα (fi (F)), ∀i ∈ J . For the product structure, we take fi = pri , where pri : j ∈J Xj −→ Xi is the projection mapping. Theorem 4.3. [29] Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Then the category sLMN -ConvTS is Cartesian closed. A pair Definition 4.4. [30]  Let L, M ∈ |Frm|, (N, ≤, ∗) ∈ |Quant| and s : L −→ M be a stratification mapping. s (X × X) satX,  = (α )α∈N is called an s-stratified LMN -uniform convergence tower space, where α ⊆ FLM isfies the following conditions: (UCT1) (UCT2) (UCT3) (UCT4) (UCT5) (UCT6) (UCT7)

[(x, x)]s ∈ α , for all x ∈ X and α ∈ N ;

∈ α whenever ≤ and ∈ α ; −1 ∈ α whenever ∈ α ; ∧ ∈ α whenever , ∈ α ; ∈ β whenever ∈ α and β ≤ α; s (X × X); ⊥N = FLM s (X × X). ∈ α , ∈ β implies ◦ ∈ α∗β for α, β ∈ N whenever ◦ ∈ FLM

    A mapping f : X,  −→ X  ,  between LMN -uniform tower convergence spaces is called uniformly continuous if (f × f )( ) ∈ α whenever ∈ α . We denote the category with objects the s-stratified LMN -uniform convergence tower spaces and morphisms the uniformly continuous mappings by sLMN -UCTS.   Definition 4.5. X,  ∈ |sLMN -UCTS| is called left-continuous if (UCTLC) ∈  A whenever A ⊆ N and ∈ α for all α ∈ A. Remark 4.6. For a left-continuous s-stratified LMN -uniform convergence tower space we define ( ) = N : ∈ α }. Then (X, ) is an s-stratified LMN -uniform convergence space defined as follows.

 {α ∈

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T.M.G. Ahsanullah, G. Jäger / Fuzzy Sets and Systems ••• (••••) •••–•••

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Definition 4.7. Let L, M, N ∈ |Frm| and s : L −→ M be a stratification mapping. Then a pair (X, ) is called an s (X × X) −→ N satisfies the following conditions: s-stratified LMN -uniform convergence space, where  : FLM (UC1) (UC2) (UC3) (UC4) (UC5)

∀x ∈ X, ([x, x]s ) = N ; ≤ implies ( ) ≤ ( ); ( ) ≤ ( −1 ); ( ) ∧ ( ) ≤  ( ∧ ); s (X × X). ( ) ∧ ( ) ≤  ( ◦ ) whenever ◦ ∈ FLM

  A mapping f : (X, ) −→ X  ,  between s-stratified LMN -uniform convergence spaces is called uniformly continuous if for all ∈ FLs (X × X), ( ) ≤  ((f × f )( )). The category of s-stratified LMN -uniform convergence spaces as objects and uniformly continuous mappings as morphisms is denoted by sLMN -UCS. If L = M = N and s = idL , then an s-stratified LMN -uniform convergence space can be identified with a stratified L-uniform convergence space studied in [24] (see also [12] for enriched lattices and [25] for frames). Conversely, given an s-stratified LMN -uniform convergence space (X, ), then the definition ∈  α ⇐⇒ ( ) ≥ α defines a left-continuous s-stratified LMN -uniform convergence tower space (X,  ). In this way, the categories sLMN -UCS and sLMN -LUCTS of left-continuous s-stratified LMN -uniform convergence tower spaces are isomorphic. 5. s-Stratified LMN -Cauchy Tower spaces and their relationship with some other s-stratified LMN -convergence tower structures Definition 5.1. Let s : L −→ M be a stratification mapping between frames L and M, and further let (N, ≤, ∗) ∈ s (X) is called an s-stratified LMN -Cauchy tower space if it |Quant|. A pair (X, c = (Cα )α∈N ), where Cα ⊆ FLM satisfies the following conditions: (CHT1) (CHT2) (CHT3) (CHT4) (CHT5)

[x]s ∈ Cα for all x ∈ X and α ∈ N ; F ≤ G, F ∈ Cα implies G ∈ Cα ; Cβ ⊆ Cα whenever α ≤ β; C⊥L = FLs (X); s (X). F ∈ Cα , G ∈ Cβ implies F ∧ G ∈ Cα∗β whenever F ∨ G ∈ FLM

 A mapping f : (X, c) −→ X  , c between s-stratified LMN -Cauchy tower spaces is called Cauchy-continuous if s (X), ∀α ∈ N , F ∈ C implies f ⇒ (F) ∈ C  . ∀F ∈ FLM α α The category of all s-stratified LMN -Cauchy tower spaces and Cauchy-continuous mappings is denoted by sLMN -ChyTS.

Example 5.2. If L = M = N = {0, 1}, we can identify s0 -stratified LMN -Cauchy tower spaces with Cauchy spaces in the sense of Fri´c and Richardson, [16] (see also, [13]). Example 5.3. If L = M = N and s = idL , then every left-continuous s-stratified LMN -Cauchy tower space can be identified with stratified L-Cauchy space due to Jäger, [26]. Example 5.4. If L = M = N and s = idL , then every s-stratified LMN -Cauchy tower space can be identified with stratified L-Cauchy space due to Boustique and Richardson, [10]. Example 5.5. If L = M = {0, 1} and N = [0, 1] equipped with ∗, then s0 -stratified LMN -Cauchy tower spaces are probabilistic Cauchy spaces due to Kent and Richardson, [35].

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We consider the underlying lattice L to be complete Boolean algebra, i.e., complete, distributive lattice with a binary operation → : L × L −→ L having double negation: (α → ⊥) → ⊥ = α, ∀α ∈ L. Using Lemma 3.1, [28],



   L s ν(x) ≤ F(ν) ≤ s ( ν(x)) → ⊥ → ⊥M (), x∈X

x∈X

s (X) and s(α) ∈ M, for we can prove that every s-stratified LM-filter is s-tight, i.e., F(αX ) = s(α), where F ∈ FLM any α ∈ L.

Lemma 5.6. Let L, M ∈ |Frm| such that the underlying lattice L is a complete Boolean algebra. Further, let s : L −→ s (X), then F is s-tight. M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . If F ∈ FLM Proof. In view of the Definition 3.1(Fs), it suffices to prove that F(αX ) ≤ s(α). Taking ν = αX in (), we have s(α) ≤ F(αX ) ≤ s(α → ⊥L ) → ⊥M . It follows that F(αX ) ∧s(α → ⊥L ) ≤ ⊥M . Using that the stratification mapping t is non-decreasing and property (SM3), we have  t (F(αX )) ∧ (α → ⊥L ) ≤ t (F(αX )) ∧ t s(α → ⊥L ) ≤ t (⊥M ) = ⊥L . Hence t (F(αX )) ≤ (α → ⊥L ) → ⊥L = α and, as also s is non-decreasing and idM ≤ s ◦ t , we deduce that F(αX ) ≤ s (t (F(αX ))) ≤ s(α), i.e., F(αX ) ≤ s(α). 2 Lemma 5.7. Let L, M ∈ |Frm| such that the underlying lattice L is a complete Boolean algebra. Let s : L −→ M and s (X). If H := F ∨ G ∈ t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . Let F, G ∈ FLM s FLM (X) is s-tight, then (F × F) ◦ (G × G) = F × G. s (X). Further, assume that H = F ∨ G ∈ F s (X) is s-tight. Then we have Proof. Let F, G ∈ FLM LM F(ρ2 ) ∧ G(σ1 ) ≤ H(ρ2 ) ∧ H(σ1 ) ≤ H(ρ

2∧ σ1 )     ≤H ( (ρ2 ∧ σ1 )(x))X (using s-tightness of H)   x∈X (†) =s x∈X (ρ2 ∧ σ1 )(x) X×X : Now for any λ ∈ L (F × F) ◦ (G × G)(λ) = ρ◦σ ≤λ (F × F)(ρ) ∧ (G × G)(σ )  (‡) = (ρ1 ×ρ2 )◦(σ1 ×σ2 )≤λ F(ρ1 ) ∧ F(ρ2 ) ∧ G(σ1 ) ∧G(σ2 )        ≤ (ρ1 ×σ2 )∧ (ρ2 ∧σ1 )(x) ≤λ F(ρ1 ) ∧ s (ρ2 ∧ σ1 )(x) ∧ G(σ2 ) (by using (†) and a result in x∈X

X×X

x∈X

Lemma  5.1, [26]) ≤ (ρ1 ×σ2 )∧ (ρ2 ∧σ1 )(x) ≤λ F(ρ1 ) ∧ s(α) ∧G(σ2 ) x∈X X×X     = (ρ1 ×σ2 )∧αX×X ≤λ F(ρ1 ∧ αX×X ) ∧ G(σ2 ) (by s-tightness)  ≤ (ρ1 ∧αX×X )×σ2 ≤λ F(ρ1 ∧ αX×X ) ∧ G(σ2 )  = η×σ2 ≤λ F(η) ∧ G(σ2 ) = F × G(λ). To prove the other part, if we take ρ2 = L X = σ1 in (‡), then it follows immediately that (‡) ≥ F × G. Hence the result follows. 2 Lemma 5.8. [24] Let L, M ∈ |Frm|, s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM s (X). Then (F ∧ G) × H = (F × H) ∧ (G × H). Similarly, H × (F ∧ G) = and t ◦ s ≥ idL . Let F, G, H ∈ FLM (H × F) ∧ H × G) holds.

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  If X,  ∈ |sLMN -UCTS|, define the s-stratified LMN -Cauchy tower structure c by F ∈ Cα ⇔ F × F ∈ α , for each α ∈ N . Lemma 5.9. Let L, M ∈ |Frm| and further assume that the underlying lattice L is a complete Boolean algebra. Let s : L −→ M and t : M −→ L be stratification mappings such that s ◦t ≥ idMand t ◦s ≥ idL . Let (N, ≤, ∗) ∈ |Quant|.    is an s-stratified LMN -Cauchy If X,  is an s-stratified LMN -uniform convergence tower space, then X, c tower space. Proof. (CHT1) Upon that [x]s × [x]s = [(x, x)]s . In fact, for any λ ∈ LX×X ,  using Lemma 3.5 [29], it follows X ([x] s × [x]s )(λ) = {[x]s (ν1 ) ∧ [x]s (ν2 ) : ν1 , ν2 ∈ L , ν1 × ν2 ≤ λ} = {s(ν1 (x)) ∧ s(ν2 (x)) : ν1 , ν2 ∈ LX , ν1 × ν2 ≤ λ} = {s (ν1 (x) ∧ ν2 (x)) : ν1 , ν2 ∈ LX , ν1 × ν2 ≤ λ} (by (SM3)) = {s ((ν1 × ν2 )(x, x)) : ν1 , ν2 ∈ LX , ν1 × ν2 ≤ λ} = {[(x, x)]s (ν1 × ν2 ) : ν1 , ν2 ∈ LX , ν1 × ν2 ≤ λ} = [(x, x)]s (λ). So, for [(x, x)]s ∈ α implies [x]s ∈ Cα . (CHT2)–(CHT4) follow easily, while for (CHT5), we let α, β ∈ N , F ∈ Cα , and G ∈ Cβ . Further, assume that F ∨ G ∈ s (X) and that it is s-tight. Then F × F ∈  and G × G ∈  . Upon using Lemma 5.7 and Definition 4.4(UCT7), FLM α β we have F × G = (F × F) ◦ (G × G) ∈ α∗β ; also, G × F ∈ α∗β , F × F ∈ α∗β and G × G ∈ α∗β . By applying Lemma 5.8 twice, one time from right and then from left on H = F ∧ G, i.e., (F ∧ G) × (F ∧ G) = [F × (F ∧ G)] ∧ [G × (F ∧ G)] = (F × F) ∧ (F × G) ∧ (G × F) ∧ (G × G), which gives us (F ∧ G) × (F ∧ G) = (F × F) ∧ (F × G) ∧ (G × F) ∧ (G × G) ∈ α∗β . Hence, we have F ∧ G ∈ Cα∗ β . 2 Lemma 5.10. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Further, Boolean assume that the underlying lattice L is a complete algebra and (N, ∗, ≤) ∈         |Quant|. Let X,  , X ,  ∈ |sLMN -UCTS|. If a mapping f : X,  −→ X ,  is uniformly continuous,    then f : X, c −→ X  , c is Cauchy-continuous. Proof. Let α ∈ N and F ∈ Cα . Then F × F ∈ α . Now one can obtain immediately by taking F = G in Proposition 3.7, [23] that f ⇒ (F) × f ⇒ (F) = (f × f )⇒ (F × F). But as f is uniformly continuous, we have  f ⇒ (F) × f ⇒ (F) = (f × f )⇒ (F × F) ∈ α . Hence f ⇒ (F) ∈ Cα . 2 Hence we have a functor ⎧ −→ sLMN-ChyTS ⎪ ⎨ sLMN-UCTS    X,  U: −→ X, c ⎪ ⎩ f −→ f   If X,  is an s-stratified LMN -uniform convergence tower space, then the s-stratified LMN -convergence tower structure q  is defined by x ∈ qα (F) ⇔ F × [x]s ∈ α ∀α ∈ N . Lemma 5.11. [30] Let L, M ∈ |Frm| and (N, ≤, ∗) ∈ |Quant|. Let s : L −→ M and t : M −→ L be stratification  mappings such that s ◦ t ≥ id and t ◦ s ≥ id . If X,  is an s-stratified LMN -uniform convergence tower space, M L  then X, q  is an s-stratified LMN -limit tower space. Lemma 5.12. [30] Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that      s ◦ t ≥ idM and t ◦ s ≥ idL . Further, assume that (N, ∗, ≤) ∈ |Quant|. Let X,  , X ,  ∈ |sLMN -UCTS|. If a       mapping f : X,  −→ X  ,  is uniformly continuous, then f : X, q  −→ X  , q  is continuous. Hence we deduce from Lemma 5.10 and 5.11 that there is a functor

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⎧ −→ ⎪ ⎨ sLMN-UCTS   V: −→ X,  ⎪ ⎩ f −→

sLMN-LimTS  X, q  f

Now we define the s-stratified LMN -limit tower structure qαc by x ∈ qαc (F) ⇔ [x]s ∧ F ∈ Cα , for any x ∈ X, α ∈ N s (X). and F ∈ FLM Lemma 5.13. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that  s ◦ t ≥ idM and t ◦ s ≥ idL ; further assume that (N, ≤, ∗) ∈ |Quant|. Let (X, c) ∈ |sLMN -ChyTS|. Then X, q c ∈ |sLMN -LimTS|.   Proof. Let (X, c) ∈ |sLMN -ChyTS|. Then it follows that X, q c is a s-stratified LMN -limit tower space. In fact, (CT2)–(CT4) are easy to check, while for (CT1) follows from the observation that [x]s = [x]s ∧ [x]s . Conditions s (X), and x ∈ q c (F) ∩ q c (G) . Then [x] ∧ F ∈ C and [x] ∧ G ∈ C imply(CT5), we take α, β ∈ N , F, G ∈ FLM s α s β α β c ing [x]s ∧ (F ∧ G) = ([x]s ∧ F) ∧ ([x]s ∧ G) ∈ Cα∗β . Hence x ∈ qα∗β (F ∧ G). 2 Lemma 5.14. Let s : L −→ M and t : M −→ L be stratification mappings between L and M such that  frames   s ◦ t ≥ idM and t ◦ s ≥ idL . Further, assume that (N, ∗, ≤) ∈ |Quant|. Let (X, c) , X , c ∈ |sLMN -ChyTS|. If a      mapping f : (X, c) −→ X  , c is Cauchy-continuous, then f : X, q c −→ X  , q c is continuous. s (X) and x ∈ q c (F). Then [x] ∧ F ∈ C , and hence by Cauchy-continuity of f , we have Proof. Let α ∈ N , F ∈ FLM s α α  f ⇒ ([x]s ∧ F) ∈ Cα . But then [f (x)]s ∧ f ⇒ (F) = f ⇒ ([x]s ∧ F) ∈ Cα . Therefore, f (x) ∈ qαc (f ⇒ (F)). 2

Hence there is a functor ⎧ ⎨ sLMN-ChyTS −→ W: −→ (X, c) ⎩ f −→

sLMN-LimTS   X, q c f

Lemma 5.15. If L is a complete Boolean algebra, M is a frame and N is a quantale, then in view of the preceding functors U, V and W, we have the following commutative diagram: sLMN-UCTS

U

−→ W◦U=V

sLMN-ChyTS W↓

sLMN-LimTS. 

s (X) and x ∈ q c (F), then F ∧ [x] ∈ C  Proof. We need to check here the equality q c = q  . For, let F ∈ FLM s α α implies (F ∧ [x]s ) × (F ∧ [x]s ) ∈ α . It follows then from Definition 4.4(UCT2) that F × [x]s ∈ α ; hence x ∈  qα (F), proving qαc (F) ⊆ qα (F). To show the opposite part, let x ∈ qα (F) implies F × [x]s ∈ α and hence (F ∧ [x]s ) × (F ∧ [x]s ) = (F × F) ∧ (F × [x]s ) ∧ ([x]s × F) ∧ ([x]s × [x]s ) ∈ α . We conclude F ∧ [x]s ∈ cα  and hence x ∈ qαc (F). 2 

6. s-Stratified LMN -convergence tower groups Definition 6.1. Let s : L −→ M and t : M −→ L be stratification mappings between the frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Let (N, ≤, ∗) ∈ |Quant| and (X, ·) be a group. Then a triple (X, ·, q = (qα )α∈N ) is called an s-stratified LMN -convergence tower group (resp. s-stratified LMN -limit tower group) if the following conditions are satisfied: (CTG1) (X, q = (qα )α∈N ) is an s-stratified LMN -convergence tower space (resp. s-stratified LMN -limit tower space);

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T.M.G. Ahsanullah, G. Jäger / Fuzzy Sets and Systems ••• (••••) •••–•••

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s (X), α, β ∈ N and x, y ∈ X; (CTM) x ∈ qα (F), y ∈ qβ (G) implies xy ∈ qα∗β (F  G), for all F, G ∈ FLM  s (X), x ∈ X and α ∈ N . (CTI) x ∈ qα (F) implies x −1 ∈ qα F −1 , for all F ∈ FLM

The category of s-stratified LMN -convergence tower groups (resp. s-stratified LMN -limit tower groups) and continuous group homomorphisms is denoted by sLMN -ConvTG (resp. sLMN -LimTG). Example 6.2. If L = M = N = {0, 1}, we can identify s0 -stratified LMN -convergence tower groups with generalized convergence groups in the classical sense, [43] (see also, [18,34]). Example 6.3. Let L = M = N and s = idL . If (X, q) satisfies the left-continuity condition (CTL) in addition to (CT1)–(CT4), then s-stratified convergence tower groups are the stratified L-convergence groups studied in [3,4]. Example 6.4. Let L = M = {0, 1} and N = [0, 1] equipped with a left-continuous t-norm ∗, then s0 -stratified LMN -convergence tower groups are probabilistic convergence groups, where s0 is the smallest stratification mapping. Note that the probabilistic convergence structure that is achieved here is the Richardson–Kent probabilistic convergence structure as studied in [45]. Example 6.5. Let L = M = {0, 1} and N = + , the set of distance distribution functions with pointwise order, and consider a sup-continuous triangle function. Then an s0 -stratified LMN -convergence tower group is a probabilistic convergence group studied in [7]. Example 6.6. Let L = M = {0, 1} and N = ([0, ∞], ≥, +) with the opposite order and ∗ = + the extended addition. Then a left-continuous s-stratified LMN -limit tower group can be identified as an +-approach limit group in the sense of Ahsanullah and Jäger [5]. If, however, ∗ = ∨(= ) is taken as in [5], then one obtains ∨-approach limit groups or so-called ultra-approach limit groups. Note that every approach limit group in the sense of Lowen and Windels [39] is an approach limit group in the sense of [5]. Example 6.7. Let L, M, N ∈ |Frm|. Let (X, ·, q) , (X, ·, p) ∈ |sLMN -ConvTG|. Then the set C(X, Y ) = {f : (X, q) s −→ (Y, p) continuous} carries a natural function space structure, [29]. For ∈ FLM (C(X, Y )), f ∈ C(X, Y ), and α ∈ N , the s-stratified LMN -convergence tower structure c on C(X, Y ) is defined by: f ∈ cαC (X,Y ) ( ) ⇔ f (x) ∈ pβ (ev( × F)) whenever x ∈ qβ (F), β ≤ α. Then it follows from Theorem 4.3 that (C(X, Y ), ·, c) ∈ |sLMN -ConvTS|. Now for f, g ∈ C(X, Y ), if we define f g(x) = f (x)g(x) and f −1 (x) = (f (x))−1 , for any C (X,Y ) x ∈ X, then one can show that f g ∈ C(X, Y ) and f −1 ∈ C(X, Y ). To prove (CTM), let f ∈ cα ( ) and C (X,Y ) g ∈ cβ ( ). Also, let x ∈ qδ (F) for δ ∈ N with δ ≤ α ∧ β. Then f (x) ∈ pδ (ev( × F)) and g(x) ∈ pδ (ev( × F)) by the continuity of the evaluation mapping ev : C(X, Y ) −→ Y , (f, x) → ev(f, x) = f (x). Thus, fg(x) = f (x)g(x) ∈ pδ∧δ (ev( × F)  ev( × F)). Hence we have f g(x) ∈ pδ (ev( × F)  ev( × F)). Since ev( ×F)  ev( ×F) ≤ ev ((  ) × F), we get f g(x) = f (x)g(x) ∈ pδ (ev ((  ) × F)) by using (CT2). It C (X,Y ) C (X,Y ) ( ). If, morefollows from the preceding definition that f g ∈ cα∧β ( ◦ ). Finally, to show (CTI), let f ∈ cα   −1 −1 over, x ∈ qβ (F), for β ≤ α, then f (x) ∈ pβ (ev( × F)). But then f (x) = (f (x)) ∈ pβ ((ev( × F))−1 .   Since (ev( × F))−1 = ev( −1 × F), we have f −1 (x) ∈ pβ (ev( −1 × F) which by definition yields that C (X,Y ) ( −1 ). This ends the proof that (C(X, Y ), ·, c) ∈ |sLMN -ConvTG|. f −1 ∈ cα Left-continuous s-stratified LMN -convergence tower groups can be identified with s-stratified LMN -convergence groups, defined as follows. Definition 6.8. Let L, M, N ∈ |Frm|, (X, ·) be a group and s : L −→ M be a stratification mapping. Then the triple (X, ·, lim) is called an s-stratified LMN -convergence group if the following conditions are satisfied: (CS) (X, lim) is an s-stratified LMN -convergence space; s (X) and x, y ∈ X; (CM) lim F(x) ∧ lim G(y) ≤ lim F  G(xy) for all F, G ∈ FLM s −1 −1 (CI) lim F(x) ≤ lim F (x ), for all F ∈ FLM (X) and x ∈ X.

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The category of all s-stratified LMN -convergence groups and continuous group homomorphisms is denoted by sLMN -ConvG. Example 6.9. If L = M = N and s = idL , then an s-stratified LMN -convergence group can be identified with the stratified L-convergence group, [3]. Proposition 6.10. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL , and let (N, ≤,∧) ∈ |Frm|. If (X, ·, q = (qα )α∈N ) is a left-continuous s-stratified LMN -convergence tower group, then X, ·, limq is an s-stratified LMN -convergence group.   Proof. That the pair X, limq is an s-stratified LMN -convergence space follows from [29]. Since  for a leftcontinuous LMN -convergence tower space (X, q), x ∈ qα (F) ⇔ limq F(x) ≥ α, where limq F(x) = {α ∈ N : x ∈ s (X) and x, y ∈ X, qα (F)}, we have for any F, G ∈ FLM G(y) lim q F(x) ∧ limq = x∈qα (F ) α ∧ y∈qβ (G ) β  = x∈qα (F ),y∈qβ (G ) α ∧ β  ≤ xy∈qα∧β (F G ) α ∧ β (by (CTM) with ∗ = ∧) = limq F  G(xy), showing that limq F(x) ∧ limq G(y) ≤ limq F  G(xy), which is (CTM), while (CTI) follows from the definition. Hence the result follows. 2 In case that (N, ≤, ∧) ∈ |Frm|, we can give the following characterization. Lemma 6.11. Let (N, ≤, ∧) ∈ |Frm|. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Let (X, ·) be a group and (X, q = (qα )α∈N ) an s-stratified LMN -convergence  tower space.Then the triple (X, ·, q = (qα )α∈N ) is an s-stratified LMN -convergence tower group if and only if m : X × X, q × q −→ (X, q), (x, y) → xy and j : (X, q) −→ (X, q), x → x −1 are continuous. s (X × X), α ∈ N and Proof. We only proof the case of m. The other part follows by definition. Let F ∈ FLM (x, y) ∈ (q × q)α (F). Then x = pr1 (x, y) ∈ qα (pr1 (F)) and y ∈ qα (pr2 (F)). Hence by (CTM), m(x, y) = xy ∈ qα∧α (m(pr1 (F) × pr2 (F))). Since pr1 (F) × pr2 (F) ≤ F , we have by (CT2), xy ∈ qα (m(F)), showing that m is s (X). Since for all α, β ∈ N , α ∧ β ≤ α, β, continuous. Conversely, let x ∈ qα (F) and y ∈ qβ (G) for F, G ∈ FLM we have by (CT3), x ∈ qα∧β (F) and y ∈ qα∧β (G). Hence (x, y) ∈ (q × q)α∧β (F × G). By continuity of m, we get xy = m(x, y) ∈ qα∧β (m(F × G)) which by using Lemma 3.6 implies that xy ∈ qα∧β (F  G). 2

Theorem 6.12. The category sLMN -ConvTG is a topological category over Grp, the category of groups, with respect to the forgetful functor. Proof. With the assistance of Lemma 3.7, and following the similar route as in the proof of Theorem 3.7, [32], one can show that the category sLMN -ConvTG is a topological category where the initial structure follows  from [29] and can be described as follows. Let (X, ·) be a group, fi : X −→ (Xi , ·i ) a group homomorphism, and Xi , ·i , q i an  , the initial s-stratified s-stratified LMN -convergence tower group. Then for a source fi : (X, ·) −→ (Xi , ·i , q i ) i∈J

LMN -convergence structure q on X is defined as: x ∈ qα (F) ⇔ fi (x) ∈ qαi (fi (F)), for all i ∈ J . By a straightforward calculation one can prove that the triple (X, ·, q) is an s-stratified LMN -convergence tower group. 2 We provide below homogeneity for s-stratified LMN -convergence tower group. Lemma 6.13. Let L, M ∈ |Frm|, s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . Let (N, ≤, ∗) ∈ |Quant|. If (X, ·, q) is an s-stratified LMN -convergence tower group, α ∈ N , F ∈ s (X) and x ∈ X, then FLM

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  x ∈ qα (F) ⇐⇒ e ∈ qα [x −1 ]s  F ⇐⇒ e ∈ qα F  [x −1 ]s

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(H )

  Proof. Let x ∈ qα (F). Then in view of (CT1), we get x −1 ∈ qN [x −1 ]s , and so, by (CTM), e = x −1 x ∈  −1  −1  −1    qN ∗α [x ]s  F , and hence e ∈ qα [x ]s  F . Conversely, if e ∈ qα [x ]s  F , then as x ∈ qN ([x]s ), by   (CTM) and Lemma 3.7, we have x = xe ∈ qN ∗α [x]s  [x −1 ]s  F = qα (F). Thus, x ∈ qα (F). The missing parts follow in a similar fashion. 2 Now using the property (H), we give necessary and sufficient conditions for an s-stratified LMN -convergence tower structure on group to be an s-stratified LMN -convergence tower group solely by the identity element of the group. Theorem 6.14. Let L, M ∈ |Frm|, s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . If (N, ∗) ∈ |Quant|, then the triple (X, ·, q = (qα )α∈N ) is an s-stratified LMN -convergence tower group if and only if the following assertions are fulfilled: (a) (b) (c) (d) (e) (f )

    x ∈ qα (F) if and only if e ∈ qα [x −1 ]s  F if and only if e ∈ qα F  [x −1 ]s ; e ∈ qα ([e]s ), for all α ∈ N ; s (X) with F ≤ G, e ∈ q (F) implies e ∈ q (G); for F, G ∈ FLM α α e ∈ qα (F) ∩ qβ (G) implies e ∈ qα∗β (F ∧ G); e ∈ qα (F) ∩ qβ (G) implies e ∈ qα∗β (F  G); e ∈ qα (F) implies e ∈ qα (F −1 ).

Proof. If the triple (X, ·, q = (qα )α∈N ) is an s-stratified LMN -convergence tower group, then (a) is precisely Lemma 6.13, while (b)–(f) follow from the definitions. To see the converse, we only prove (CTM).   Let x ∈ qα(F) and s (X) and α, β ∈ N . Then by (a), e ∈ q [x −1 ]  F and e ∈ q G  [y −1 ] . Hence y ∈ qβ (G), for F, G ∈ FLM α s β s     by (e), we have e = ee ∈ qα∗β ([x −1 ]s  F)  (G  [y −1 ]s ) . Then again by (a), x ∈ qα∗β (F  G)  [y −1 ]s   and using (a) once more in conjunction with Lemma 3.7, we have e ∈ qα∗β (F  G)  [y −1 ]s  [x −1 ]s ) =   qα∗β (F  G)  ([(xy)−1 ]s ) , whence xy ∈ qα∗β (F  G). 2 7. s-Stratified LMN -uniformizability of s-stratified LMN -convergence tower groups Let (X, ·, q = (qα )α∈N ) be an s-stratified LMN -convergence tower group. We define a mapping, ωl : X × X −→ X, (x, y) → x −1 y. Lemma 7.1. Let L, M ∈ |Frm|, s: L −→ M and t : M −→ L be s-stratification mappings such that s ◦ t ≥ idM and  t ◦ s ≥ idL . Let (X, ·) and X  , · be groups with identity element e and e respectively. If f : X −→ X is a group s (X) and , ∈ F s (X × X), the followings are satisfied: homomorphism, then for x ∈ X, F, G ∈ FLM LM (a) ωl⇒ ([(x, x)]s ) = [e]s ; (b) ωl⇒ (F × G) = F −1  G;  −1 (c) ωl⇒ ( −1 ) = ωl⇒ ( ) ; ⇒ ≤ ωl⇒ ( ◦ ); (d) ωl⇒ ( )  ⇒ ωl ( )⇒ ⇒ ωl ( ) = ωl ((f × f )⇒ ( )). (e) f Proof. (a) Let ν ∈ LX and x ∈ X. Then ωl⇒ ([(x, x)]s )(ν) = [(x, x)]s (ν ◦ ωl ) = s(ν ◦ ωl )(x, x) = s(ν(ωl (x, x)) = s(ν(e)) = [e]s (ν). (b) This follows the same way as in Lemma 3.6. (c) Let ν ∈ LX and ∈ LsLM (X × X). Then we have

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ωl⇒ ( −1 )(ν) = −1 (ν ◦ ωl )([ν ◦ ωl ∈ LX×X ])   −1  (ν). = (ν ◦ ωl )−1 = (ν −1 ◦ ωl ) = ωl⇒ ( ) s X (d) Let ν ∈ L and , ∈ FLM (X × X) such that ◦ exists. We have [ωl⇒ ( )  ωl⇒ ( )](ν)  = ν1 ν2 ≤ν [ωl⇒ ( )(ν1 ) ∧ ωl⇒ ( )(ν2 )]  [ (ν1 ◦ ωl ) ∧ (ν2 ◦ ωl )] ≤ =  ν1 ν2 ≤ν [ (λ) ∧ (γ )] λ◦γ  =(ν1 ◦ωl )◦(ν2 ◦ωl )≤(ν1 ν2 )◦ωl ≤ν◦ωl ≤ λ◦γ ≤ν◦ωl [ (λ) ∧ (γ )] = ◦ (ν ◦ ωl ) = ωl⇒ ( ◦ )(ν). In fact, for any (z1 , z2 ) ∈ X × X, and λ, γ ∈ LX×X , we have λ ◦γ (z1 , z2 ) = (ν1 ◦ ωl ) ◦ (ν2 ◦ ωl )(z1 , z2 ) = z∈X ν1 ◦ ωl (z1 , z) ∧ ν2 ◦ ωl (z, z2 )  = z∈X ν1 (z1−1 z) ∧ ν2 (z−1 z2 ) = ab=z−1 zz−1 z2 ν1 (a) ∧ ν2 (b) 1  = ab=z−1 z2 ν1 (a) ∧ ν2 (b) 1

= ν1  ν2 (z1−1 z2 ) = (ν1  ν2 ) ◦ ω(z1 , z2 ), which yields that λ ◦ γ ≤ ν ◦ ωl . s (X × X) and ν ∈ LX . Then (e) Let ∈ FLM  f ⇒ ωl⇒ ( ) (ν) = ωl⇒ ( ) (f ← (ν)) = (f ← (ν) ◦ ωl ) = ((ν ◦ ωl ) ◦ (f × f )) = (f × f )⇒ ( )(ν ◦ ωl ) = ωl⇒ ((f × f )⇒ ( )) (ν). In fact, for any (x, y) ∈ X × X, we have f ← (ν) ◦ ωl (x, y) = f ← (ν)(x −1 y) = ν(f (x −1 y)) = ν(f (x)−1 f (y)) = ν(ωl (f (x), f (y)) = ν ◦ ωl (f (x), f (y)) = (ν ◦ ωl ) ◦ (f × f )(x, y).

2

If (X, ·, q = (qα )α∈N ) is an s-stratified LMN -convergence tower group, then the s-stratified LMN  we define  q uniform convergence tower structure q on X by ∈ α if and only if e ∈ qα ωl⇒ ( ) . Theorem 7.2. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such that assume that (N, ≤, ∗) ∈ |Quant|. If (X, ·, q = (qα )α∈N ) is an s-stratified s ◦ t ≥ idM and t ◦ s ≥ idL . Furthermore,   LMN -limit tower group, then X, q is an s-stratified LMN -uniform convergence tower space.   q Proof. (UCT1) Since e ∈ qα ([e]s ), we have qα ([e]s ) = qα ωl⇒ ([(x, x)]s ) , by Lemma 7.1(a). Hence [(x, x)]s ∈ α , for any x ∈ X and α ∈ N . (UCT2)–(UCT6) follow easily with the help of Lemma 7.1. We only prove  (UCT7).  For, let α, β ∈ N ,  ∈ α and s (X × X). Then e ∈ q ω⇒ ( ) and e ∈ q ω⇒ ( ) , implying assume that

◦

∈ F

∈ β . Furthermore, α β LM l    l that e = ee ∈ qα∗β ωl⇒ ( )  ωl⇒ ( ) , which by Lemma 7.1(d) implies that e ∈ qα∗β ωl⇒ ( ◦ ) . Hence ◦ ∈ q α∗β . 2 Proposition 7.3. Let L, M, N ∈ |Frm|. Let s : L −→ M and t : M −→ L be stratification mappings such that s ◦ t ≥ idM and t ◦ s ≥ idL . If (X, ·, q = (qα )α∈N ) is a left-continuous s-stratified LMN -limit tower group, then X, ·, limq is an s-stratified LMN -limit group, and hence it induces an s-stratified LMN -uniform convergence space. Proof. The first part of this proof follows from Proposition 6.10, for the LMN -uniformizality, we remark that if s (X × X) −→ N and define for any ∈ we replace the mapping in the Theorem 5.6, [4] by the mapping U l : FLM s (X × X) by FLM  s U l ( ) = {lim F(e) : F ∈ FLM (X), F ≤ ωl⇒ ( )},

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and use the fact that U l ([(x, x)]s ) =

15



s {lim F(e) : F ∈ FLM (X), F ≤ ωl⇒ ([(x, x)]s )} = lim[e]s (e) = N

then following the similar route as in the  proof ofthe Theorem 5.6, [4] for frames, we can see that every left-continuous s-stratified LMN -limit tower group X, ·, limq is an s-stratified LMN -uniform convergence space. We leave the details for the interested reader. 2 8. s-Stratified LMN -Cauchy tower groups and their relationship with s-stratified LMN -limit tower groups Definition 8.1. Let s : L −→ M and t : M −→ L be stratification mappings between the frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Further assume that (N, ≤, ∗) ∈ |Quant|. If (X, ·) is group, and (X, c) an s-stratified LMN -Cauchy tower space, then the triple (X, ·, c) is called an s-stratified LMN -Cauchy tower group if the following conditions are satisfied: s (X) and ∀α, β ∈ N ; (CHTM) F ∈ Cα , G ∈ Cβ implies F  G ∈ Cα∗β , ∀F, G ∈ FLM s (X) and ∀α ∈ N . (CHTI) F ∈ Cα implies F −1 ∈ Cα , ∀F ∈ FLM

The category of all s-stratified LMN -Cauchy tower groups and Cauchy-continuous group homomorphisms is denoted by sLMN -ChyTG. Example 8.2. If L = M = N = {0, 1}, we can identify s0 -stratified LMN -Cauchy tower groups with Cauchy groups in the sense of [17]. Example 8.3. If L = M = N and s = idL , then left-continuous s-stratified LMN -Cauchy tower groups are stratified L-Cauchy groups that arose from stratified L-Cauchy rings studied in Ahsanullah, [2]. Lemma 8.4. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such  that s ◦ t ≥ idM and t ◦ s ≥ idL ; further assume that (N, ≤, ∗) ∈ |Quant|. If (X, ·, c) ∈ |sLMN -ChyTG|, then X, ·, q c ∈ |sLMN -LimTG|. Proof. In view of Lemma 5.13, we only prove (CTM) and (CTI). s (X) with α, β ∈ N . If x ∈ q c (F) and y ∈ q c (G), then [x] ∧ F ∈ C (CTM) Let x, y ∈ X and F, G ∈ FLM s α α β and [y]s ∧ G ∈ Cβ implying ([x]s ∧ F)  ([y]s ∧ G) ∈ Cα∗β by (CHTM). Since in view of Lemma 3.7(g), c (([x]s ∧ F)  ([y]s ∧ G)) = ([xy]s ∧ (F  G)), we have [xy]s ∧ (F  G) ∈ Cα∗β . But then xy ∈ qα∗β (F  G). s c (CTI) Let F ∈ FLM (X), x ∈ X and α ∈ N . If now x ∈ qα (F), then [x]s ∧ F ∈ Cα . But then by Lemma 3.7(b), (e) and (CHTI), [x −1 ]s ∧ F −1 = ([x]s ∧ F)−1 ∈ Cα and hence, x −1 ∈ qαc (F −1 ). 2 Definition 8.5. Let s : L −→ M and t : M −→ L be stratification mappings between the frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Moreover, assume that (N, ≤, ∧) ∈ |Frm|. Then an s-stratified LMN -convergence tower group (X, ·, q) is called strongly normal if s (X), α, β ∈ N, e ∈ q (F −1  F), e ∈ q (F  F −1 ) and e ∈ q (G) ∀F , G ∈ FLM α  β α ⇒ e ∈ qα∧β F  G  F −1 .

  Given (X, ·, q) ∈ |sLMN -LimTG|, define sLMN -Cauchy tower space X, cq by  F ∈ Cαq ⇔ e ∈ qα F  F −1   or simply by F ∈ Cα ⇐⇒ e ∈ qα F  F −1 , if there is no danger of confusion. Lemma 8.6. Let s : L −→ M and t : M −→ L be stratification mappings between frames L and M such  that s ◦ t ≥ idM and t ◦ s ≥ idL . Moreover, assume that (N, ≤, ∧) ∈ |Frm|. If (X, ·, q) ∈ |sLMN -LimTG|, then X, ·, cq is an

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s-stratified LMN -Cauchy tower group if and only if (X, ·, q) is a strongly normal s-stratified LMN -convergence tower group. Proof. Define s (X), α ∈ N. F ∈ Cα ⇐⇒ e ∈ qα (F −1  F), e ∈ qα (F  F −1 ), and e ∈ qβ (G), ∀F , G ∈ FLM   Assume that (X, ·, q) is a strongly normal s-stratified LMN -convergence tower group. We prove X, ·, cq ∈ |sLMN -ChyTG|. (CHT1) Since in view of Theorem 6.14(a), x ∈ qα ([x]s ) implies e ∈ qα ([x −1 ]s  [x]s ), and e ∈ qα ([x]s  [x −1 ]s ). Hence [x]s ∈ Cα . (CHT2) Let F ≤ G, α ∈ N and F ∈ Cα . Then e ∈ qα (F −1  F). Because of F −1  F ≤ G −1  G, e ∈ qα (G −1  G). Hence G ∈ Cα . (CHT3) Let α, β ∈ N with α ≤ β. If F ∈ Cβ , then e ∈ qβ (F −1  F), and hence e ∈ qα (F −1  F). This implies F ∈ Cα . (CHT4) Clearly true. (CHT5) Let α, β ∈ N , F ∈ Cα and G ∈ Cβ ; further assume that F ∨ G exists.  Then e ∈ qα (F −1  F), e ∈ qα (F  F −1 ), e ∈ qβ (G −1  G) and e ∈ qβ (G  G −1 ). So, e ∈ qα∧β (F −1  F) ∧      (G −1  G) . But since F −1  F ∧ G −1  G = (F −1 ∧ G −1 )  (F ∧ G)  = (F ∧ G)−1  (F ∧ G ,   we have e ∈ qα∧β (F ∧ G)−1  (F ∧ G) . Hence F ∧ G ∈ Cα∧β . (CHTM) Let F ∈ Cα , G ∈ Cβ , and α, β ∈ N . Then e∈ qα (F −1  F), e ∈ qα (F  F −1 ) and e ∈ qβ (G  G −1 ). Then by strong normality, e ∈ qα∧β F  G  G −1  F −1 . Now since F  G  (F  G)−1 = F  G  G −1  F −1 ,   we have e ∈ qα∧β (F  G)  (F  G)−1 . Hence F  G ∈ Cα∧β . F−1 ). (CHTI) Let α ∈ N and F ∈Cα . Then  e−1∈ q−1α (F −1 −1 −1 Then e ∈ qα (F  F ) = qα (F )  F . This implies F −1 ∈ Cα . s (X), α, β ∈ N, e ∈ To prove the converse, let (X, ·, c) be an s-stratified LMN -Cauchy tower group. Let F, G ∈ FLM  qα (F −1 F), e ∈ qα (F F −1 ) and e ∈ qβ (G). Show that e ∈ qα∧β F  G  F −1 . Then since by definition, F ∈ Cα   and G ∈ Cβ , we have F  G ∈ Cα∧β . This implies e ∈ qα∧β (F  G)  (F  G)−1 . There is no harm to assume 3.7(a), we have F  G  G −1  F −1 ≤ F  [e]s  F −1 ≤ F  G  F −1 and, hence [e]s ≤ G,then using Lemma  −1 e ∈ qα∧β F  G  F . 2 Let sLMN -SNLimTG denote the category whose objects are strongly normal s-stratified LMN -limit tower groups, and morphisms are continuous group homomorphisms. Lemma 8.7. Let s : L −→ M and t : M −→ L be stratification mappings between  frames L and M such that s ◦ t ≥ idM and t ◦ s ≥ idL . Moreover, assume that (N, ≤, ∧) ∈ |Frm|. Let (X, ·, q) , X  , ·, q  ∈ |sLMN -SNLimTG|. If f : X −→ X  is a group homomorphism, then the following assertions are equivalent:  (a) f : (X, q) −→ X  , q  is continuous;     (b) f : X, cq −→ X  , cq is Cauchy-continuous.     q Proof. (⇒): Let F ∈ Cα . Then e ∈ qα F −1  F which implies e = f (e) ∈ qα (f ⇒ (F))−1  f ⇒ (F) . Similarly,   e ∈ qα f ⇒ (F)  (f ⇒ (F))−1 . q

Hence f ⇒ (F) ∈ Cα . s (X). If now x ∈ q (F), then (⇐): Let x ∈ X, α ∈ N and F ∈ FLM α     q −1 −1 −1 x ∈ qα (F ). So, e = x x ∈ qα F −1  F , and also e ∈ qα F  F −1 , which implies that F ∈ Cα and hence q

q

by Cauchy-continuity of f , f ⇒ (F) ∈ Cα . Thus, we have f ⇒ (F) ∧ [f (x)]s ∈ Cα .

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  This gives, e ∈ qα (f ⇒ (F) ∧ [f (x)]s )−1  (f ⇒ (F) ∧ [f (x)]s ) and e ∈ qα ((f ⇒ (F ∧ [f (x)]s )  (f ⇒ (F) ∧  [f (x)]s )−1 . But it follows from Lemma 3.7 that   ⇒ (f (F) ∧ [f (x)]s )−1  (f ⇒ (F) ∧ [f (x)]s )    ⇒ = (f ⇒ (F))−1 ∧ ([f (x)]s )−1 )  (f ⇒ (F) ∧ [f (x)]s ) ≤ [f (x)]−1 s  f (F) , and so,    e ∈ qα [(f (x))−1 ]s  f ⇒ (F) . Similarly, e ∈ qα f ⇒ (F)  [(f (x))−1 ]s . Hence f (x) ∈ qα (f ⇒ (F)). 2 Hence we have functors ⎧ ⎨ sLMN-SNLimTG F: (X, q) ⎩ f and

−→ −→ −→

⎧ ⎨ sLMN-ChyTG −→ G: −→ (X, c) ⎩ f −→

sLMN-ChyTG   X, cq f

sLMN-SNLimTG   X, qc f

Theorem 8.8. sLMN -ChyTG is isomorphic to sLMN -SNLimTG. Proof. In view of Lemma 8.6 and Lemma 8.7, we only need to show that G ◦ F = idsLMN −SNLimTG and F ◦ G = idsLMN −ChyTG . Let (X, q) ∈ |sLMN − SNLimTG|. Then we have G ◦ F (X, q) = G (F(X, q)) =    q q s (X) and x ∈ G X, cq = X, q c = (X, q) = idsLMN −SNLimTG (X, q), whence q c = q. Indeed, let F ∈ FLM     q q qαc (F). Then F ∧ [x]s ∈ Cα and so, e ∈ qα (F ∧ [x]s )  (F ∧ [x]s )−1 . Now as (F ∧ [x]s )  (F ∧ [x]s )−1 =     −1 ∧ [x]−1 ) ≤ (F  [x]−1 ), we get (F ∧ [x]s )  (F −1 ∧ [x]−1 s ) (by Lemma 3.7(b)) and as (F ∧ [x]s )  (F s s q −1 e ∈ qα (F  [x]s ) (by Definition 4.1(CT2)), which by Lemma 6.13 imply x ∈ qα (F) proving that qαc (F) ⊆ qα (F). Next, to show the opposite part, we let x ∈ qα (F). Then we have the following: e ∈ qα (F  [x]−1 s ) and as x −1 ∈ qα (F −1 ), e = xx −1 ∈ qα (F  F −1 ); similarly, e = xx −1 ∈ qα ([x]s  F −1 ), and also e ∈ qα ([e]s ). But we have (F ∧ [x]s )  (F ∧ [x]s )−1 = (F ∧ [x]s )  (F −1 ∧ [x]−1 s ) −1 ∧ [x]−1 )] (by applying Lemma 3.7(d)) = [F  (F −1 ∧ [x]−1 )] ∧ [[x] s  (F s s −1 −1 = (F  F ) ∧ (F  [x]s ) ∧ ([x]s  F −1 ) ∧ ([x]s  [x]−1 s ) (again by applying Lemma 3.7(d)) −1 ) ∧ [e] (using Lemma 3.7(f)). ) ∧ ([x]  F = (F  F −1 ) ∧ (F  [x]−1 s s s  q Upon using Definition 4.1(CT5), we get e ∈ qα∧α (F ∧ [x]s )  (F ∧ [x]s )−1 , which yields that F ∧ [x]s ∈ Cα and qc

hence x ∈ qαc (F), proving that qα (F) ⊆ qαc (F). On the other hand, since F ∈ Cα ⇔ e ∈ qαc (F  F −1 ) ⇔ F ∈ Cα   qc c −1 qc c and F ∈ Cα ⇔ e ∈ qα (F  F) ⇔ F ∈ Cα ; we have c = c, proving that F ◦ G (X, c) = F X, q =  also, q

X, cq

c

q

= (X, c) = idsLMN −ChyTG (X, c), for any (X, c) ∈ |sLMN − ChyTG|.

2

Corollary 8.9. If the underlying group is Abelian, then sLMN -ChyTG is isomorphic to sLMN -LimTG Proof. This goes almost the same way as in preceding theorem with taking into account the Theorem 6.14, and the fact that for Abelian group, stratified LMN -limit tower groups are strongly normal. 2 9. Conclusions Upon using the notions of so-called stratification mapping between frames with some additional conditions and s-stratified LM-filters, we introduced and studied the notion of s-stratified LMN -convergence tower groups, s-stratified LMN -limit tower groups and s-stratified LMN -Cauchy tower groups. Considering the notion of s-stratified LMN -uniform convergence tower spaces studied in [30] we are able to show that every s-stratified LMN -limit tower group gives rise to a s-stratified LMN -uniform convergence tower space. It would however be

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interesting to study the completion of s-stratified LMN -Cauchy tower group as a generalization of {0, 1}-completion of Cauchy group undertaken in [17] (see also [8]). We look into this problem in one of our future works. Acknowledgement We are sincerely thankful to the reviewers for carefully reading our earlier manuscript and providing various suggestions which have greatly helped to improve the present version. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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