M -fuzzifying gated amalgamations of M -fuzzifying geometric interval spaces

Abstract

In M-fuzzifying interval spaces, the notion of M-fuzzifying base-point orders is introduced, by which some characterizations of M-fuzzifying geometric (resp. Peano, Pasch) interval spaces are obtained. Then notions of M-fuzzifying gated sets, M-fuzzifying gate maps and M-fuzzifying gated amalgamations are introduced. It is shown that M-fuzzifying gated sets are preserved by M-fuzzifying IP-surjective mappings, and that the M-fuzzifying Peano (resp. Pasch, modular, JHC) property is preserved by M-fuzzifying gated amalgamations. In particular, the M-fuzzifying sand-glass property is also preserved by M-fuzzifying gated amalgamations provided that the corresponding M-fuzzifying gate maps are M-fuzzifying II-mappings.

1 Introduction

Theory of interval spaces is an important part of the theory of convex structures. In fact, interval operators derived from mathematical structures, such as graphic theory [2], poset [6], median algebra [8], metric space [10], lattice [18] and vector space [19], not only display some unified geometric properties, but also provide a natural and frequent method of describing or constructing convex structures [1 , 19]. In addition, interval operators, equipped by some special properties, such as geometric property, modular property, Peano-Pasch property and sand-glass property, connect and enhance many features of convex structures [19].

Convexity has been extended into fuzzy settings by several means. The notion of fuzzy real convex numbers in fuzzy metric spaces was defined and studied by Tripathy and Das [3 , 17]. Also, the notion of fuzzy convexities defined by Rosa [11] was further extended into M-convexities by Maruyama [7]. Actually, either a fuzzy convexity or an M-convexity is a crisp family of fuzzy sets or M-fuzzy sets satisfying certain set of axioms. However, from a totally different point of view, Shi and Xiu introduced M-fuzzifying convexities where each subset of the underling set can be regarded as a convex set to some degree [16]. They also showed that an M-fuzzifying closure structure is an M-fuzzifying convex structure iff its closure operator is domain finite. Latter, Shi and Xiu introduced the notion of M-fuzzifying interval spaces, and proved that an M-fuzzifying convex structure is generated by an M-fuzzifying interval space iff it is of arity ≤2 [25]. Also, Shi and Li introduced the notion of M-fuzzifying restricted hull spaces and obtained a one-to-one correspondence between M-fuzzifying convex structures and M-fuzzifying restricted hull spaces [15]. Based on these results, Wu and Bai further introduced several notions including M-fuzzifying JHC convex structures [22], M-fuzzifying geometric (resp. modular, Peano, Pasch, sand-glass) interval spaces [23] and M-fuzzifying convex matroids [24], and showed that these notions together with M-fuzzifying convex structures, ‘arity ≤2’ are closely related.

In this paper, we define the notion of M-fuzzifying base-point orders and characterize M-fuzzifying geometric (resp. Peano, Pasch) interval spaces. Then we introduce notions of M-fuzzifying gated maps and M-fuzzifying gated amalgamations. We find that M-fuzzifying gated amalgamations preserve M-fuzzifying geometric (resp. Peano, Pasch, Modular, JHC) property, and M-fuzzifying sand-glass property provided that their corresponding M-fuzzifying gate maps are M-fuzzifying II-functions.

2 Preliminaries

Throughout, X and Y are nonempty sets. We denote 2^X = {A : A ⊆ X} and $2_{2}^{X} = {{x, y} : x, y \in X}$ .

M is a completely distributive lattice with an inverse involution ^′, whose least and largest elements are denoted by ⊥ and ⊤. An element a ∈ M is called a prime if for all b, c ∈ M, b ∧ c ≤ a implies b ≤ a or c ≤ a. The set of all primes in M ∖ {⊤} is denoted by P (M). The set {a′ : a ∈ P (M)} is denoted by J (M). For each a ∈ M, there exist φ ⊆ P (M) and ψ ∈ J (M) such that a = ⋀ φ = ⋁ ψ [21]. For all p, q ∈ M, p ≤ q iff p ≰ r implies q ≰ r for all r ∈ P (M) [23] iff s ≤ p implies s ≤ q for all s ∈ J (M) [16].

A binary relation ≺ on M is defined by: for all a, b ∈ M, a ≺ b iff for each φ ⊆ M, b ≤ ⋁ φ always implies the existence of d ∈ φ such that a ≤ d. For a ∈ M, β (a) = {b : b ≺ a} [16, 20]. For all p, q ∈ M, p ≤ q iff s ≺ p implies s ≤ q for all s ∈ β (⊤) [16, 23]. The implication operator → : X × X → M is defined by: a → b = ⋁ {c ∈ M : a ∧ c ≤ b} for all a, b ∈ X [5].

The set of all M-fuzzy sets on X is denoted by M^X. We denote U_[r] = {x ∈ X : U (x) ≥ r} and U^(r) = {x ∈ X : U (x) ≰ r} for U ∈ M^X and r ∈ M [12].

Let f : X → Y be a mapping. The M-fuzzy mapping $f_{M}^{\to} : M^{X} \to M^{Y}$ is defined by: $f_{M}^{\to} (U) (y) = ⋁ {U (x) : f (x) = y}$ for U ∈ M^X and y ∈ Y. Also, $f_{M}^{ł} : M^{Y} \to M^{X}$ is defined by: $f_{M}^{ł} (V) (x) = V (f (x))$ for V ∈ M^Y and x ∈ X [13 , 20].

Definition 2.1. [16] A mapping $C : 2^{X} \to M$ is called an M-fuzzifying convexity and $(X, C)$ is called an M-fuzzifying convex structure, if $C$ satisfies

$C (\emptyset) = C (X) = ⊤$ ;

If {U_i} _i∈Ω ⊆ 2^X, then $C (⋂_{i \in Ω} U_{i}) \geq ⋀_{i \in Ω} C (U_{i})$ ;

If {U_i} _i∈Ω ⊆ 2^X is totally ordered by inclusion, then $C (⋃_{i \in Ω} U_{i}) \geq ⋀_{i \in Ω} C (U_{i})$ .

Theorem 2.2. [16] The M-fuzzifying hull operator ${co}_{C} : 2^{X} \to M^{X}$ (briefly, co) of an M-fuzzifying convex structure $(X, C)$ is defined by: $\begin{matrix} \forall A \in 2^{X}, \forall x \in X, co (A) (x) & = & ⋀_{x \notin B \supseteq A} (C (B))^{'} . \end{matrix}$

Then for all A ∈ 2^X and x ∈ X,

co (∅) (x) = ⊥;

co (A) (x) =⊤ whenever x ∈ A;

co (A) (x) = ⋀ _x∉B⊇A ⋁ _y∉Bco (B) (y);

$co (A) (x) = ⋁_{F \in 2_{fin}^{A}} co (F) (x)$ .

Conversely, if co : 2^X → M^X satisfies (MCO1)-(MFD), then the mapping $C_{co} : 2^{X} \to M$ , defined by: $\begin{matrix} \forall A \in 2^{X}, C_{co} (A) = ⋀_{x \notin A} (co (A) (x))^{'} . \end{matrix}$ is an M-fuzzifying convexity with ${co}_{C_{co}} = co$ . The restriction co^seg of co on $2_{2}^{X}$ is called the segment operator of $(X, C)$ , which is still denoted by co.

Definition 2.3. [22, 25] An M-fuzzifying convex structure $(X, C)$ is of arity ≤n ( $n \in ℕ$ ), if $\begin{matrix} \forall A \in 2^{X}, C (A) & = & ⋀_{x \notin A} ⋀_{F \in 2_{fin}^{A}, | F | \leq n} (co (F) (x))^{'} . \end{matrix}$

Definition 2.4. [15] An operator $H : 2_{fin}^{X} \to M^{X}$ is called an M-fuzzifying restricted hull operator, if for all $F, G \in 2_{fin}^{X}$ and x ∈ X,

$H (\emptyset) (x) = ⊥$ ;

$H (F) (x) = ⊤$ whenever x ∈ F;

$H (G) (x) \land ⋀_{y \in G} H (F) (y) \leq H (F) (x)$ .

Theorem 2.5. [15] The restriction co_fin of an M-fuzzifying hull operator co on $2_{fin}^{X}$ is an M-fuzzifying restricted hull operator, which is still denoted by co.

Definition 2.6. [22] An M-fuzzifying convexity $C$ on X is called an M-fuzzifying JHC convexity, if co (A) (z) = ⋁ _x∈Xco ({a, x}) (z) ∧ co (A) (x) for all a, z ∈ X and A ∈ 2^X ∖ {∅}.

Definition 2.7. [25] $I : X \times X \to M^{X}$ is called an M-fuzzifying interval operator and $(X, I)$ is called an M-fuzzifying interval space, if for all x, y ∈ X,

$I (a, b) (a) = I (a, b) (b) = ⊤$ ;

$I (a, b) = I (b, a)$ .

Further, $(X, I)$ is called an M-fuzzifying geometric interval space [23], if for all a, b, c, d, z ∈ X,

$I (a, a) = a_{⊤}$ ;

$I (a, b) (c) \land I (a, c) (d) \leq I (a, b) (d)$ ;

$I (a, b) (c) \land I (a, b) (d) \land I (a, d) (c) \leq I (b, c) (d)$ .

Finally, an M-fuzzifying geometric interval space $(X, I)$ is called an M-fuzzifying modular space, if $⋁_{x \in X} M (a, b, c) (x) = ⋁_{x \in X} (I (a, b) \land I (b, c) \land I (a, c)) (x) = ⊤$ for all a, b, c ∈ X [23].

Theorem 2.8. [23] An operator $I : X \times X \to M^{X}$ satisfying (MGI1) is an M-fuzzifying geometric interval operator iff it satisfies (MGI2&3) below.

$I (a, b) (c) \land I (b, c) (d) \leq I (a, b) (d) \land I (a, d) (c)$ for all a, b, c, d ∈ X.

In fact, $I$ is an M-fuzzifying geometric interval operator iff $I^{(r)}$ is a geometric interval operator for all r ∈ P (M), where $I^{(r)} (a, b) = (I (a, b))^{(r)}$ [22].

Let $(X, I_{X})$ and $(Y, I_{Y})$ be M-fuzzifying interval spaces. A mapping f : X → Y is called an M-fuzzifying IP-mapping (resp. II-mapping), if $f_{M}^{\to} (I_{X} (a, b)) \leq I_{Y} (f (a), f (b))$ (resp. $f_{M}^{\to} (I_{X} (a, b)) = I_{Y} (f (a), f (b))$ ) for all a, b ∈ X [25].

Theorem 2.9. [25] If $(X, C)$ is an M-fuzzifying convex structure, then the restriction of co on $2_{2}^{X}$ is an M-fuzzifying interval operator generated by $C$ . Conversely, if $(X, I)$ is an M-fuzzifying interval space and $C_{I} : 2^{X} \to M$ is defined by: $\begin{matrix} \forall A \in 2^{X}, C_{I} (A) & = & ⋀_{z \notin A} ⋀_{x, y \in A} (I (x, y) (z))^{'}, \end{matrix}$ then $C_{I}$ is an M-fuzzifying convexity generated by $C$ . In addition, $I \leq {co}_{C_{I}}^{seg}$ .

Theorem 2.10. [25] An M-fuzzifying convexity is generated by an M-fuzzifying interval operator iff it is of arity ≤2 iff it is generated by its segment operator.

Definition 2.11. [22, 23] An M-fuzzifying interval space $(X, I)$ is called an

M-fuzzifying Peano interval space, if $I (b, c) (y) \land I (a, y) (z) \leq ⋁_{x \in X} I (a, b) (x) \land I (c, x) (z)$ for a, b, c, y, z ∈ X;

M-fuzzifying Pasch interval space, if $I (p, a) (\bar{a}) \land I (p, b) (\bar{b}) \leq ⋁_{x \in X} I (a, \bar{b}) (x) \land I (b, \bar{a}) (x)$ for $p, a, \bar{a}, \bar{b} \in X$ ;

M-fuzzifying sand-glass interval space, if $I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \leq ⋁_{v \in X} I (v, w) (p) \land I (c, d) (w)$ for a, b, c, d, p, v ∈ X.

If $(X, I)$ is an M-fuzzifying Peano (resp. Pasch, sand-glass) interval space, then we say $I$ has M-fuzzifying Peano (resp. Pasch, sand-glass) property. If the segment operator of an M-fuzzifying convexity has M-fuzzifying Peano (resp. Pasch, sand-glass) property, then the convexity is called an M-fuzzifying Peano (resp. Pasch, sand-glass) convexity.

Definition 2.12. [22] Let $(X, I)$ be an M-fuzzifying interval space, a, b, c ∈ X and U, V ∈ M^X. Define $[a (bc)]_{I}, (UV)_{I} \in M^{X}$ by: for all z ∈ X,

$[a (bc)]_{I} (z) = ⋁_{x \in X} I (b, c) (x) \land I (a, x) (z)$ ;

$(UV)_{I} (z) = ⋁_{a, b \in X} U (a) \land V (b) \land I (a, b) (z)$ ; in particular, we denote $U ∣_{I}$ for $(UU)_{I}$ .

Theorem 2.13. [22] An M-fuzzifying interval space $(X, I)$ is an M-fuzzifying Peano interval space iff $[a (bc)]_{I} = [(ab) c]_{I}$ for all a, b, c ∈ X.

Theorem 2.14. [22] An M-fuzzifying JHC convex structure is of arity ≤2. In addition, an M-fuzzifying convexity $C$ of arity ≤2 is an M-fuzzifying JHC convexity iff ${co}_{C}^{seg}$ has M-fuzzifying Peano property.

Theorem 2.15. [22] Let $(X, C)$ be an M-fuzzifying convex structure and r ∈ P (M). Then the operator co^(r) : 2^X → 2^X, defined by: co^(r) (A) = (co (A)) ^(r) for all A ∈ 2^X, is the hull operator of $(X, C_{[r^{'}]})$ .

Theorem 2.16. [23] If $(X, I)$ is an M-fuzzifying interval space and r ∈ P (M), then $(C_{I})_{[r^{'}]} = C_{I^{(r)}}$ . In addition, $U ∣_{I} \leq U$ iff $U^{(s)} \in C_{I^{(s)}}$ for all U ∈ M^X and s ∈ P (M).

Definition 2.17. [26] $B : X \times X \to M$ is called an M-fuzzifying quasi-partial order, if for all x, y, z ∈ X,

$B (x, x) = ⊤$ ;

$B (x, z) \land B (z, y) \leq B (x, y)$ .

B

is further called an M-fuzzifying partial order, if

$B (x, y) \land B (y, x) = ⊥$ whenever x ≠ y.

3 M-fuzzifying base-point orders and M-fuzzifying gated sets

Definition 3.1. Let $(X, I)$ be an M-fuzzifying interval space and b ∈ X. A mapping $B_{I}^{b} : X \times X \to M$ , defined by: $B_{I}^{b} (x, y) = ⋀_{z \in X} I (b, x) (z) \to I (b, y) (z)$ for all x, y ∈ X, is called the M-fuzzifying base-point at b (with respect to $(X, I)$ ).

Theorem 3.2. If $(X, I)$ is an M-fuzzifying interval space and b ∈ X, then $B_{I}^{b}$ is an M-fuzzifying quasi-order. If $I (b, x) (y) \land I (b, y) (x) \neq ⊥$ implies x = y, then $B_{I}^{b}$ is an M-fuzzifying partial order. If $I$ satisfies (MGI2), then $B_{I}^{b} (x, y) = I (b, y) (x)$ .

Theorem 3.3. Let $(X, I)$ be an M-fuzzifying interval space. Then the following statements are equivalents.

$(X, I)$ is an M-fuzzifying geometric interval space.

$B_{I}^{b}$ is an M-fuzzifying partial order, $B^{b} (b, y) = ⊤$ and $B_{I}^{b} (u, v) \land I (b, c) (u) \land I (b, c) (v) = B_{I}^{c} (v, u) \land I (b, c) (u) \land I (b, c) (v)$ for all b, c, u, v, y ∈ X.

$B_{I}^{b}$ is an M-fuzzifying partial order, $B^{b} (b, y) = ⊤$ and $I (u, v) (x) \land B_{I}^{b} (u, v) = B_{I}^{b} (u, x) \land B_{I}^{b} (x, v) \land B_{I}^{b} (u, v)$ for all b, u, v, x, y ∈ X.

Proof. (1) ⇒ (2). $B_{I}^{b} (b, x) = I (b, x) (b) = ⊤$ . Also, $B_{I}^{b} (x, y) \land B_{I}^{b} (y, x) \leq y_{⊤} (x)$ by Theorem 3.2, (MGI2&3) and (MGI1). Thus $B_{I}^{b}$ is an M-fuzzifying partial order. Further, it is follows from (MGI3) that $\begin{matrix} B_{I}^{b} (u, v) \land I (b, c) (u) \land I (b, c) (v) \leq B_{I}^{c} (v, u) . \end{matrix}$

Hence the inequality from left to right is clear. The inverse inequality is similar. Therefore (2) holds.

(2) ⇒ (3). We firstly check that (MGI2) is fulfilled. If (MGI2) fails, then there exist a, b, c, d ∈ X such that $I (a, b) (c) \land I (a, c) (d) ≰ I (a, b) (d)$ . So there exists $r \in β (I (a, b) (c) \land I (a, c) (d))$ and $r ≰ I (a, b) (d)$ .

Since $B_{I}^{b} (b, c) = ⊤$ and $I (a, b) (b) = ⊤$ , we have $\begin{matrix} r & \leq & B_{I}^{b} (b, c) \land I (a, b) (c) \land I (a, b) (b) \\ = & B_{I}^{a} (c, b) \land I (a, b) (c) \land I (a, b) (b) \\ \leq & B_{I}^{a} (c, b) \leq I (a, c) (d) \to I (a, b) (d) . \end{matrix}$

It implies that $r = I (a, c) (d) \land r \leq I (a, b) (d)$ which is a contradiction. Therefore (MGI2) holds.

Now, we check (3). $B_{I}^{b} (u, v) \land I (u, v) (x) \leq I (b, v) (x)$ by (MGI2). Thus $\begin{matrix} B_{I}^{b} (u, v) \land I (u, v) (x) \\ = I (b, v) (x) \land I (b, v) (u) \land I (u, v) (x) \\ = B_{I}^{v} (x, u) \land I (b, v) (x) \land I (b, v) (u) \\ = B_{I}^{b} (u, x) \land I (b, v) (u) \land I (b, v) (x) . \end{matrix}$

This shows $I (u, v) (x) \land B_{I}^{b} (u, v) \leq B_{I}^{b} (u, x) \land B_{I}^{b} (x, v) \land B_{I}^{b} (u, v)$ . Conversely, we have $\begin{matrix} B_{I}^{b} (u, x) \land B_{I}^{b} (x, v) \land B_{I}^{b} (u, v) \\ = B_{I}^{b} (u, x) \land I (b, v) (u) \land I (b, v) (x) \land B_{I}^{b} (u, v) \\ = B_{I}^{v} (x, u) \land I (b, v) (u) \land I (b, v) (x) \land B_{I}^{b} (u, v) \\ \leq B_{I}^{v} (x, u) \land B_{I}^{b} (u, v) = I (u, v) (x) \land B_{I}^{b} (u, v) . \end{matrix}$

Therefore (3) holds.

(3) ⇒ (1). Suppose that (MGI2) fails. There exist a, b, c, d ∈ X and $r \in β (I (a, b) (c) \land I (a, c) (d))$ such that $r ≰ I (a, b) (d)$ . Thus $r \leq I (a, b) (c) = I (a, b) (c) \land B_{I}^{a} (a, b) = B_{I}^{a} (a, c) \land B_{I}^{a} (c, b) \land B_{I}^{a} (a, b) = B_{I}^{a} (c, b)$ . But $B_{I}^{a} (c, b) \leq I (a, c) (d) \to I (a, b) (d) ≱ r$ , a contradiction. Thus (MGI2) holds.

Now, we check that (MGI1) and (MGI3).

(MGI1). If x ∈ X and x ≠ a, then $I (a, a) (x) = B_{I}^{b} (a, x) \land B_{I}^{b} (x, a) \land B_{I}^{b} (a, a) = B_{I}^{b} (a, x) \land B_{I}^{b} (x, a)$ .

(MGI3). Let a, b, c, d ∈ X. Since (MGI2) implies $B^{x} (u, v) = I (x, v) (u)$ for all x, u, v ∈ X, we have $\begin{matrix} I (a, b) (c) \land I (a, b) (d) \land I (a, d) (c) \\ = B_{I}^{a} (c, b) \land B_{I}^{a} (d, b) \land B_{I}^{a} (c, d) \\ = I (c, b) (d) \land B_{I}^{a} (c, b) \leq I (c, b) (d) . \end{matrix}$

Definition 3.4. Let $(X, I)$ be an M-fuzzifying interval space, b ∈ X and U ∈ M^X. We define $↓_{I}^{b} (U), ↑_{I}^{b} (U) \in M^{X}$ as follows.

∀z ∈ X, $↓_{I}^{b} (U) (z) = ⋁_{x \in X} B_{I}^{b} (z, x) \land U (x)$ .

∀z ∈ X, $↑_{I}^{b} (U) (z) = ⋁_{x \in X} B_{I}^{b} (x, z) \land U (x)$ .

Theorem 3.5. Let $(X, C)$ be an M-fuzzifying convex structure.

If $C$ is of arity ≤2, then its has M-fuzzifying Peano property iff $↓_{co}^{b} (co (A)) ∣_{co} \leq ↓_{co}^{b} (co (A))$ for all b ∈ X and A ∈ 2^X.

$C$ has M-fuzzifying Pasch property iff $↑_{co}^{b} (co (A)) ∣_{co} \leq ↑_{co}^{b} (co (A))$ for all b ∈ X and A ∈ 2^X.

If $C$ is an M-fuzzifying sand-glass convexity, then $↑_{co}^{b} (p_{⊤}) ∣_{co} \leq ↑_{co}^{b} (p_{⊤})$ for all b, p ∈ X.

Proof. (1) Necessity. Let x, y, z ∈ X. If A =∅, then $↓_{co}^{b} (co (A)) ∣_{co} = ↓_{co}^{b} (co (A)) = \underline{⊥}$ . If A≠ ∅, then $co ({b} \cup A) (u) = ↓_{co}^{b} (co (A)) (u)$ for all b, u ∈ X by M-fuzzifying JHC property and (MGI2). Thus

$\begin{matrix} ↓_{co}^{b} (co (A)) ∣_{co} (z) \\ = ⋁_{u, v \in X} co ({b} \cup A) (u) \land co ({b} \cup A) (v) \\ \land co ({u, v}) (z) \leq co ({b} \cup A) (z) \\ = ↓_{co}^{b} (co (A)) (z) . \end{matrix}$

Therefore $↓_{co}^{b} (co (A)) ∣_{co} \leq ↓_{co}^{b} (co (A))$ .

Sufficiency. Let a, b, c ∈ X and A = {a, c}. Clearly, $[b (ac)]_{co} = ↓_{co}^{b} (co (A)) \leq co ({a, b, c})$ by (MGI2) and (MRH3). Since $↓_{co}^{b} (co (A)) ∣_{co} \leq ↓_{co}^{b} (co (A))$ and $a, b, c \in (↓_{co}^{b} (co (A)))^{(r)}$ for all r ∈ P (M), we have $co ({a, b, c}) \leq ↓_{co}^{b} (co (A))$ by Theorems 2.15 and 2.16. Hence $[b (ac)]_{co} = co ({a, b, c}) = ↓_{co}^{c} (co ({a, b})) = [c (ab)]_{co}$ . Thus $C$ is an M-fuzzifying Peano convexity.

(2) Necessity. Let A ∈ 2^X. Then

$\begin{matrix} ↑_{co}^{b} (co (A)) ∣_{co} (z) \\ = ⋁_{x, y, u, v \in X} (co ({b, y}) (v) \land co ({x, y}) (z)) \\ \land co ({b, x}) (u) \land co (A) (u) \land co (A) (v) \\ \leq ⋁_{x, u, v \in X} (⋁_{w \in X} co ({b, z}) (w) \land co ({x, v}) (w)) \\ \land co ({b, x}) (u) \land co (A) (u) \land co (A) (v) \\ = ⋁_{x, u, v, w \in X} (co ({b, x}) (u) \land co ({x, v}) (w)) \\ \land co ({b, z}) (w) \land co (A) (u) \land co (A) (v) \\ \leq ⋁_{u, v, w \in X} (⋁_{p \in X} co ({b, w}) (p) \land co ({u, v}) (p)) \\ \land co ({b, z}) (w) \land co (A) (u) \land co (A) (v) \\ \leq ⋁_{w, p \in X} co ({b, w}) (p) \land co ({b, z}) (w) \land co (A) (p) \\ \leq ⋁_{p \in X} co ({b, z}) (p) \land co (A) (p) = ↑_{co}^{b} (co (A)) (z) . \end{matrix}$

Sufficiency. Let $p, a, b, \bar{a}, \bar{b} \in X$ and $A = {\bar{a}, b}$ . We have $↑_{co}^{a} (co (A)) (\bar{a}) = ↑_{co}^{a} (co (A)) (b) = ⊤$ and $↑_{co}^{a} (A) (p) \geq co ({a, p}) (\bar{a})$ . Thus, by hypothesis, $\begin{matrix} ⋁_{x \in X} co ({a, \bar{b}}) (x) \land co (A) (x) \\ = ↑_{co}^{a} (co (A)) (\bar{b}) \geq ↑_{co}^{a} (co (A)) ∣_{co} (\bar{b}) \\ \geq ↑_{co}^{a} (co (A)) (b) \land ↑_{co}^{a} (co (A)) (p) \land co ({b, p}) (\bar{b}) \\ \geq co ({a, p}) (\bar{a}) \land co ({b, p}) (\bar{b}) . \end{matrix}$

(3) Let b, p, z ∈ X. By (MGI2), (MRH3) and the M-fuzzifying sand-glass property, we have $\begin{matrix} ↑_{co}^{b} (p_{⊤}) ∣_{co} (z) \\ = ⋁_{x, y \in X} B_{co}^{b} (p, x) \land B_{co}^{b} (p, y) \land co ({x, y}) (z) \\ \leq ⋁_{w \in X} co ({b}) (w) \land co ({w, z}) (p) \leq ↑_{co}^{b} (p_{⊤}) (z) . \end{matrix}$

Therefore $↑_{co}^{b} (p_{⊤}) ∣_{co} \leq ↑_{co}^{b} (p_{⊤})$ .

Definition 3.6. Let $(X, I)$ be an M-fuzzifying geometric interval space, A ∈ 2^X and b ∈ X. An point a ∈ A is called an M-fuzzifying gate of b in A, denoted by a ⊢ _Ab, if $I (b, x) (a) = ⊤$ for all x ∈ A.

Proposition 3.7. Let $(X, I)$ be an M-fuzzifying geometric interval space, A ⊆ X, a, b ∈ A and c ∈ X.

If a ⊢ _Ab, then a = b.

If a ⊢ _Ac and b ⊢ _Ac, then a = b.

a ⊢ _Ac iff $B_{I}^{c} (a, x) = ⊤$ for all x ∈ A.

Definition 3.8. Let $(X, I)$ be an M-fuzzifying geometric interval space and A ⊆ X.

A is called an M-fuzzifying gated set if each point of X has an M-fuzzifying gate in A.

If A is an M-fuzzifying gated set, then a map p_X : X → A, assigning each point to its M-fuzzifying gate, is called the M-fuzzifying gated map of A. p_X is conveniently denoted by p.

Proposition 3.9. If $(X, I)$ is an M-fuzzifying geometric interval space and A ∈ 2^X is an M-fuzzifying gated set, then $I (a, b) (z) = I (p (a), b) (z)$ for all a ∈ X and b, z ∈ A.

Theorem 3.10. Let $(X, I_{X})$ and $(Y, I_{Y})$ be M-fuzzifying geometric interval spaces and f : X → Y be an M-fuzzifying IP-surjective mapping. If A ∈ 2^X is an M-fuzzifying gated set, then f (A) ∈2^Y is an M-fuzzifying gated set. In addition, if p_X : X → A and p_Y : Y → f (A) are M-fuzzifying gated maps, then f (p_X (x)) = p_Y (f (x)) for all x ∈ X.

4 M-fuzzifying gated amalgamations

Theorem 4.1. Let $(X_{i}, I_{i})$ (i = 1, 2) be M-fuzzifying geometric interval spaces, X₁ ∩ X₂ = Y and X₁ ∪ X₂ = X such that

Y is an M-fuzzifying gated set of both X_i;

$I_{1} (a, b) = I_{2} (a, b) \in M^{Y}$ for all a, b ∈ Y.

Then there exists an M-fuzzifying geometric interval operator $I$ on X subjecting to (MA1)-(MA3).

$I$ extends $I_{i}$ . I.e., if a, b ∈ X_i, z ∈ X, then $\begin{matrix} I (a, b) (z) = {\begin{matrix} I_{i} (a, b) (z), & z \in X_{i}, \\ ⊥, & z \notin X_{i}; \end{matrix} \end{matrix}$

if a ∈ X₁ and b ∈ X₂, then there exists y ∈ Y such that $I (a, b) (y) = ⊤$ ;

if p_i : X_i → Y (i = 1, 2) are M-fuzzifying gate maps, a ∈ X₁, b ∈ X₂ and z ∈ X, then $\begin{matrix} I (a, b) (z) = {\begin{matrix} I_{1} (a, p_{2} (b)) (z), & z \in X_{1}, \\ I_{2} (p_{1} (a), b) (z), & z \in X_{2} . \end{matrix} \end{matrix}$

In addition, $I$ also satisfies (MA4).

X₁ and X₂ are M-fuzzifying gated sets, and there exist M-fuzzifying gate maps P_i : X → X_i extending p_j (i, j = 1, 2, i ≠ j), respectively.

Finally, $I$ is unique with respect to (MA1) and (MA2).

Proof. We prove it by the following four steps.

Step 1. (MA1) and (MA3) are compatible.

Let a ∈ X₁, b ∈ Y and z ∈ X. If z ∈ X₁, then $I (a, b) (z) = I_{1} (a, b) (z)$ by (MA1). Also, $I (a, b) (z) = I_{1} (a, p_{2} (b)) (z) = I_{1} (a, b) (z)$ by Proposition 3.7(1) and (MA3). If z ∈ X₂ ∖ X₁, then $I (a, b) (z) = ⊥$ by (MA1), and $I (a, b) (z) = I_{2} (p_{1} (a), b) (z) = I_{1} (p_{1} (a), b) (z) = ⊥$ by (MA3) and (2). Thus (MA1) and (MA2) are compatible.

Step 2. Clearly, $I$ , determined by (MA1) and (MA3), is an M-fuzzifying interval operator.

(MGI1) clearly follows from (MA1). We show that (MG2&3) holds for $I$ . That is, for all a, b, u, v ∈ X, $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ \leq I (a, b) (u) \land I (u, b) (v) . \end{matrix}$ We prove this by the following cases.

(1) a, b ∈ X₁. We have the following two subcases.

(a) If u, v ∈ X₁, then (MG2&3) holds by (MA1).

(b) If v ∉ X₁, then $I (a, b) (v) = ⊥$ . If v ∈ X₁ and u ∉ X₁, then $I (a, v) (u) = ⊤$ . Thus (MG2&3) holds.

(2) a, b ∈ X₂. Similar to (1).

(3) a ∈ X₁ and b ∈ X₂.

(a) v ∈ X₁ and u ∉ X₁. Then $I (a, v) (u) = ⊥$ and (MG2&3) immediately follows.

(b) v, u ∈ X₁. By (MGI2&3) of $I_{1}$ and (MA2), $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ = I_{1} (a, p_{2} (b)) (v) \land I_{1} (a, v) (u) \\ \leq I_{1} (a, p_{2} (b)) (u) \land I_{1} (u, p_{2} (b)) (v) \\ = I (a, b) (u) \land I (u, b) (v) . \end{matrix}$

If u ∈ X₂ ∖ X₁, then by (MGI2&3) of $I_{2}$ , we have $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ = I_{2} (p_{1} (a), b) (v) \land I_{2} (p_{1} (a), v) (u) \\ \leq I_{2} (p_{1} (a), b) (u) \land I_{2} (u, b) (v) \\ = I (a, b) (u) \land I (u, b) (v) . \end{matrix}$

If u ∈ X₁ ∖ X₂, then $\begin{matrix} I (a, v) (u) = I_{1} (a, p_{2} (v)) (u) . \end{matrix}$ (c1)

By $I_{1} (u, p_{2} (v)) (p_{1} (u)) = ⊤$ and (MG2&3) of $I_{1}$ , $\begin{matrix} I (a, v) (u) & \leq I_{1} (a, p_{2} (v)) (p_{1} (u)) \\ = I_{1} (p_{1} (a), p_{2} (v)) (p_{1} (u)) . \end{matrix}$ (c2)

Since $I_{2} (p_{1} (a), v) (p_{2} (v)) = ⊤$ , by (MG2&3) of $I_{2}$ , $\begin{matrix} I (a, b) (v) & = I_{2} (p_{1} (a), b) (v) \\ \leq I_{2} (p_{2} (v), b) (v) \land I_{2} (p_{1} (a), b) (p_{2} (v)) . \end{matrix}$ (c3)

By Proposition 3.9 and (MGI2) of $I_{1}$ , we have $\begin{matrix} I_{2} (p_{1} (a), b) (p_{2} (v)) & = I_{1} (p_{1} (a), p_{2} (b)) (p_{2} (v)) \\ \leq I_{1} (a, p_{2} (b)) (p_{2} (v)) . \end{matrix}$ (c4)

By (MG2&3) of $I_{2}$ , we have $\begin{matrix} I_{2} (p_{1} (a), b) (p_{2} (v)) & \land I_{2} (p_{1} (a), p_{2} (v)) (p_{1} (u)) \\ \leq I_{2} (p_{1} (u), b) (p_{2} (v)); \end{matrix}$ (c5) $\begin{matrix} I_{2} (p_{1} (u), b) (p_{2} (v)) & \land I_{2} (p_{2} (v), b) (v) \\ \leq I_{2} (p_{1} (u), b) (v) \\ = I (u, b) (v) . \end{matrix}$ (c6)

Similarly, by (MG2&3) of $I_{1}$ , we have $\begin{matrix} I_{1} (a, p_{2} (b)) (p_{2} (v)) \land I_{1} (a, p_{2} (v)) (u) \\ \leq I_{1} (a, p_{2} (b)) (u) = I (a, b) (u) . \end{matrix}$ (c7)

By (c1), (c2), ··· and (c7), (MGI2&3) holds for $I$ .

If u ∈ Y, then (MGI2&3) follows from $I_{2}$ . Thus $I$ is an M-fuzzifying geometric interval operator.

Step 3. Let P_i : X → X_i be defined by: $\begin{matrix} P_{i} (z) = {\begin{matrix} z, & z \in X_{i}, \\ p_{j} (z), & z \in X_{j}, \end{matrix} (i, j = 1, 2, i \neq j) . \end{matrix}$

Then X_i is an M-fuzzifying gated set with respect to the M-fuzzifying gate map P_i for each i = 1, 2. In addition, P_i ∣ _{X
_j} = p_j, showing that P_i extends p_j for all i, j = 1, 2 with i ≠ j. That is, (MA4) holds.

Step 4. We show that $I$ is unique.

Let $(X, \bar{I})$ be an M-fuzzifying geometric interval space satisfying (MA1) and (MA2). By (MA1), $\bar{I} (a, b) = I (a, b)$ for all a, b ∈ X_i (i = 1, 2).

Now, let a ∈ X₁, b ∈ X₂ and z ∈ X.

By (MA2), there exists y ∈ Y such that $\bar{I} (a, b) (y) = ⊤$ . Since p₁ (a) is the M-fuzzifying gate of a with respect to $(X_{1}, I_{1})$ , we have $\bar{I} (a, y) (p_{1} (a)) = I_{1} (a, y) (p_{1} (a)) = ⊤$ . Thus $⊤ = \bar{I} (a, b) (y) \land \bar{I} (a, y) (p_{1} (a)) \leq \bar{I} (a, b) (p_{1} (a))$ . This shows p₁ (a) is also the M-fuzzifying gate of a in X₂ with respect to $(X, \bar{I})$ . Similarly, p₂ (b) is the M-fuzzifying gate of b in X₁ with respect to $(X, \bar{I})$ .

Finally, we prove that $\bar{I} (a, b) (z) = I (a, b) (z)$ .

If z ∈ X₁, then $I (a, b) (z) = I_{1} (a, p_{2} (b)) (z) = \bar{I} (a, p_{2} (b)) (z) \leq \bar{I} (a, b) (z)$ by (MA1) and (MGI2). Since $\bar{I} (z, b) (p_{2} (b)) = ⊤$ , we have $\bar{I} (a, b) (z) \leq \bar{I} (a, p_{2} (b)) (z) = I_{1} (a, p_{2} (b)) (z) = I (a, b) (z)$ by (MGI2&3) and (MA1). Thus $I (a, b) (z) = \bar{I} (a, b) (z)$ for all z ∈ X₁. Similarly, if z ∈ X₂, then $I (a, b) (z) = I_{2} (p_{1} (a), b) (z) = \bar{I} (p_{1} (a), b) (z) \leq \bar{I} (a, b) (z)$ . Since $\bar{I} (a, z) (p_{1} (a)) = ⊤$ , we have $\bar{I} (a, b) (z) \leq \bar{I} (p_{1} (a), b) (z) = I_{2} (p_{1} (a), b) (z) = I (a, b) (z)$ by (MGI3). Therefore $I (a, b) (z) = \bar{I} (a, b) (z)$ for all z ∈ X₂. In conclusion, we have $\bar{I} = I$ .

For convenience, the conditions (1), (2), (MA1) and (MA2) in Theorem 4.1 together is called the M-fuzzifying amalgamation condition. In addition, the M-fuzzifying geometric operator $I$ is called the M-fuzzifying gated amalgamation of $I_{i}$ (i = 1, 2).

Corollary 4.2. If $(X_{i}, I_{i})$ (i = 1, 2) are M-fuzzifying geometric interval spaces fulfilling the M-fuzzifying amalgamation condition, then their M-fuzzifying gated amalgamation is an M-fuzzifying modular operator iff both $I_{1}$ and $I_{2}$ are M-fuzzifying modular operators.

Theorem 4.3. Let $(X_{i}, I_{i})$ (i = 1, 2) be M-fuzzifying geometric interval spaces fulfilling the M-fuzzifying amalgamation conditions, and $I$ be their M-fuzzifying gated amalgamation. Then

$I$ has M-fuzzifying Peano property iff both $I_{1}$ and $I_{2}$ have M-fuzzifying Peano property;

$I$ has M-fuzzifying Pasch property iff both $I_{1}$ and $I_{2}$ have M-fuzzifying Pasch property.

Proof. The necessity directly follows from (MA1). Thus we only need to check the sufficiencies.

(1) Let a, b, c, x, y ∈ X. We want to prove that $\begin{matrix} I (a, b) (x) \land I (c, x) (y) \\ \leq ⋁_{z \in X} I (a, c) (z) \land I (b, z) (y) . \end{matrix}$ (*)

By the symmetric role of X₁ and X₂, we only need to consider the following cases and subcases.

(i) a, b, c ∈ X₁ ∖ X₂.

If x ∉ X₁ or y ∉ X₁, then (*) holds trivially. If x, y ∈ X₁, then (*) follows from (MA1).

(ii) a ∈ X₂ and b, c ∈ X₁.

(iia) x, y ∈ X₁. Then p₂ (a) ⊢ _{X
₁}a relative to $(X, I)$ . Consider p₂ (a) , b, c, x, y ∈ X₁. We have $\begin{matrix} I (b, a) (x) \land I (x, c) (y) \\ \leq ⋁_{z \in X_{1}} I_{1} (c, p_{2} (a)) (z) \land I_{1} (z, b) (y) \\ \leq ⋁_{z \in X} I (c, a) (z) \land I (z, b) (y) . \end{matrix}$

(iib) x ∈ X₂ and y ∈ X₁. Then p₂ (x) ⊢ _{X
₁}x relative to $(X, I)$ . Thus $I (b, a) (x) \leq I (b, a) (p_{2} (x))$ and $I (c, x) (y) \leq I (c, p_{2} (x)) (y)$ by (MGI2) and (MGI2&3). Since p₂ (x) , y ∈ X₁, we have, by (iia), $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq I (b, a) (p_{2} (x)) \land I (c, p_{2} (x)) (y) \\ \leq ⋁_{z \in X} I (a, c) (z) \land I (b, z) (y) . \end{matrix}$

(iic) x, y ∈ X₂. Then p₁ (b) ⊢ _{X
₂}b and p₁ (c) ⊢ _{X
₂}c relative to $(X, I)$ . Thus $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq ⋁_{z \in X_{2}} I_{2} (p_{1} (c), a) (z) \land I_{2} (p_{1} (b), z) (y) \\ \leq ⋁_{z \in X} I (a, c) (z) \land I (b, z) (y) . \end{matrix}$

(iii) a, b ∈ X₂ and c ∈ X₁. If x ∈ X₁ ∖ X₂, then $I (a, b) (x) = ⊥$ and (*) is trivial.

(iiia) x, y ∈ X₂. Then p₁ (c) ⊢ _{X
₂}c relative to $(X, I)$ . Thus $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ = I_{2} (b, a) (x) \land I_{2} (p_{1} (c), x) (y) \\ \leq ⋁_{z \in X_{2}} I_{2} (a, p_{1} (c)) (z) \land I_{2} (b, z) (y) \\ \leq ⋁_{z \in X} I (c, a) (z) \land I (b, z) (y) . \end{matrix}$

(iiib) x ∈ X₂ and y ∈ X₁. Then p₁ (y) ⊢ _{X
₂}y relative to $(X, I)$ . Thus $I (c, x) (y) \leq I (c, x) (p_{1} (y)) \land I (c, p_{1} (y)) (y)$ by (MGI2&3). Consider the configurations a, b, c, x, p₁ (y). Similar to (iiia), we obtain that $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq (I (b, a) (x) \land I (c, x) (p_{1} (y))) \\ \land I (c, p_{1} (y)) (y) \\ \leq ⋁_{z \in X} I (c, a) (z) \land I (b, z) (p_{1} (y)) \\ \land I (c, p_{1} (y)) (y) . \end{matrix}$

If z ∈ X₂ ∖ X₁, then p₂ (z) ⊢ _{X
₁}z relative to $(X, I)$ . Thus, by (MGI2) and (MGI2&3), we have $\begin{matrix} I (c, a) (z) \land I (b, z) (p_{1} (y)) \\ \leq I (c, a) (p_{2} (z)) \land I (b, p_{2} (z)) (p_{1} (y)) \end{matrix}$

Notice that p₂ (z) ∈ X₁ for all z ∈ X₂. We obtain that $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq ⋁_{z \in X_{1}} I (c, a) (z) \land I (b, z) (p_{1} (y)) \\ \land I (c, p_{1} (y)) (y) . \end{matrix}$

For z ∈ X₁, we have z, c, p₁ (y) , y ∈ X₁ and b ∈ X₂. Applying (iva) with the labels 1 and 2 interchanged to the configurations z, b, c, p₁ (y) , y, we have $\begin{matrix} I (z, b) (p_{1} (y)) \land I (c, p_{1} (y)) (y) \\ \leq ⋁_{u \in X} I (z, c) (u) \land I (b, u) (y) . \end{matrix}$

Hence by (MGI2), we have $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq ⋁_{z \in X_{1}} ⋁_{u \in X} I (c, a) (z) \land I (z, c) (u) \land I (b, u) (y) \\ \leq ⋁_{u \in X} I (c, a) (u) \land I (b, u) (y) . \end{matrix}$

(iv) a, c ∈ X₂ and b ∈ X₁.

(iva) x, y ∈ X₂. Then a, p₁ (b) , c, x, y ∈ X₂. Thus $\begin{matrix} I (a, b) (x) \land I (c, x) (y) \\ = I_{2} (a, p_{1} (b)) (x) \land I_{2} (c, x) (y) \\ \leq ⋁_{z \in X_{2}} I_{2} (c, a) (z) \land I_{2} (p_{1} (b), z) (y) \\ \leq ⋁_{z \in X} I (c, a) (z) \land I (b, z) (y) . \end{matrix}$

(ivb) x ∈ X₁ and y ∈ X₂. Then p₁ (x) ⊢ _{X
₂}x relative to $(X, I)$ . Notice that p₁ (x) , y ∈ X₂. By (MGI2), (MA3) and (iva), we have $\begin{matrix} I (a, b) (x) \land I (c, x) (y) \\ \leq I (a, b) (p_{1} (x)) \land I (c, p_{1} (x)) (y) \\ \leq ⋁_{z \in X} I (c, a) (z) \land I (b, z) (y) . \end{matrix}$

(ivc) x, y ∈ X₁. Then p₂ (a) ⊢ _{X
₁}a relative to $(X, I)$ . Thus $I (b, a) (x) \leq I (b, p_{2} (a)) (x)$ by (MGI2&3). Now, interchange the labels 1 and 2. We have p₂ (a) , b, x, y ∈ X₂ and c ∈ X₁. By (iiia) and (MA1), $\begin{matrix} I (p_{2} (a), b) (x) \land I (c, x) (y) \\ \leq ⋁_{u \in X} I (p_{2} (a), c) (u) \land I (b, u) (y) \\ = ⋁_{u \in X_{1}} I (p_{2} (a), c) (u) \land I (b, u) (y) . \end{matrix}$

So, if u ∈ X₁, then b ∈ X₂ and a, c, p₂ (a) , u ∈ X₁. By symmetrical role of $I$ on X₁ and X₂ and by (iva), $\begin{matrix} I (c, p_{2} (a)) (u) & = & I (b, a) (p_{2} (a)) \land I (c, p_{2} (a)) (u) \\ \leq & ⋁_{z \in X} I (a, c) (z) \land I (z, b) (u) . \end{matrix}$

Hence, by (MGI2), we have $\begin{matrix} I (b, a) (x) \land I (c, x) (y) \\ \leq I (p_{2} (a), b) (x) \land I (c, x) (y) \\ \leq ⋁_{u \in X_{1}} I (p_{2} (a), c) (u) \land I (b, u) (y) \\ \leq ⋁_{u \in X_{1}} ⋁_{z \in X} I (a, c) (z) \land I (z, b) (u) \land I (b, u) (y) \\ \leq ⋁_{z \in X} I (a, c) (z) \land I (z, b) (y) . \end{matrix}$

Therefore (*) holds.

(2) We need to prove that for all a, b, c, x, y ∈ X, $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$ (**)

We prove this by following three cases.

(i) a, b, c ∈ X₁. Then (**) holds trivially.

(ii) a ∈ X₂ and b, c ∈ X₁.

(iia) x, y ∈ X₁. So p₂ (a) ⊢ _{X
₁}a relative to $(X, I)$ . Thus $I (b, a) (x) \leq I (b, p_{2} (a)) (x)$ and $I (b, a) (y) \leq I (b, p_{2} (a)) (y)$ by (MGI2&3). Hence by (i), we have $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ \leq I (p_{2} (a), b) (x) \land I (p_{2} (a), c) (y) \\ \leq ⋁_{z \in X_{1}} I (b, y) (z) \land I (c, x) (z) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$

(iib) x ∈ X₁ and y ∈ X₂. Then p₂ (y) ⊢ _{X
₁}y relative to $(X, I)$ . Thus $I (b, p_{2} (y)) (z) \leq I (b, y) (z)$ and $I (a, c) (y) \leq I (a, p_{2} (y)) (y)$ by (MGI2). Consider the configurations a, b, c, x, p₂ (y). By (iia), $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ \leq I (a, b) (x) \land I (a, c) (p_{2} (y)) \\ \leq ⋁_{z \in X} I (b, p_{2} (y)) (z) \land I (c, x) (z) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$

(iic) x, y ∈ X₂ and z ∈ X. Then p₁ (b) ⊢ _{X
₂}b and p₁ (c) ⊢ _{X
₂}c relative to $(X, I)$ . Thus $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ = I_{2} (a, p_{1} (b)) (x) \land I_{2} (a, p_{1} (c)) (y) \\ \leq ⋁_{z \in X_{2}} I_{2} (p_{1} (b), y) (z) \land I_{2} (p_{1} (c), x) (z) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$

(iii) a, b ∈ X₂ and c ∈ X₁. If x ∈ X₁ ∖ X₂, then $I (a, b) (x) = ⊥$ and (**) is trivial.

(iiia) x, y ∈ X₂. Then p₁ (c) ⊢ _{X
₂}c and $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ = I_{2} (a, b) (x) \land I_{2} (a, p_{1} (c)) (y) \\ \leq ⋁_{z \in X_{2}} I_{2} (b, y) (z) \land I_{2} (p_{1} (c), x) (z) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$

(iiib) y ∈ X₁ and x ∈ X₂. Then p₁ (y) ⊢ _{X
₂}y and p₂ (a) ⊢ _{X
₁}a relative to $(X, I)$ . Thus $I (a, c) (y) = I_{1} (p_{2} (a), c) (y) \leq I_{1} (p_{2} (a), c) (p_{1} (y)) = I_{2} (a, p_{1} (c)) (p_{1} (y))$ by (MA3) and (MGI2). Hence we have $\begin{matrix} I (a, b) (x) \land I (a, c) (y) \\ \leq I_{2} (a, b) (x) \land I_{2} (a, p_{1} (c)) (p_{1} (y)) \\ \leq ⋁_{z \in X_{2}} I_{2} (b, p_{1} (y)) (z) \land I_{2} (p_{1} (c), x) (z) \\ \leq ⋁_{z \in X} I (b, y) (z) \land I (c, x) (z) . \end{matrix}$

Corollary 4.4. Let $(X_{i}, C_{i})$ (i = 1, 2) be M-fuzzifying convex structures of arity ≤2. If their segment operators co_i are M-fuzzifying geometric interval operators fulfilling the M-fuzzifying amalgamation condition, and co is the M-fuzzifying gated amalgamation of co_i, then $C_{co}$ is an M-fuzzifying JHC convexity iff $C_{i}$ are M-fuzzifying JHC convexities.

Proof. It follows from Theorems 2.9 and 2.10 that $C_{i}$ is generated by its segment operator ${co}_{i}^{seg}$ .

By Theorem 2.14 and Theorem 4.3(1), $C_{co}$ is an M-fuzzifying JHC convexity iff co has M-fuzzifying Peano property iff co_i have M-fuzzifying Peano property iff $C_{i}$ have M-fuzzifying JHC property.

Theorem 4.5. Let $(X_{i}, I_{i})$ (i = 1, 2) be M-fuzzifying geometric interval spaces having M-fuzzifying amalgamation conditions. If $I_{i} (p_{i} (a), p_{i} (b)) (w) \leq (p_{i})_{M}^{\to} (I_{i} (a, b)) (w)$ for all a, b ∈ X_i and w ∈ X₁ ∩ X₂ = Y, then their M-fuzzifying gated amalgamation $I$ has M-fuzzifying sand-glass property iff both $I_{1}$ and $I_{2}$ have M-fuzzifying sand-glass property.

Proof. The necessity is clear.

Sufficiency. Let a, b, c, d, p, v ∈ X. We prove that $\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ \leq ⋁_{w \in X} I (c, d) (w) \land I (v, w) (p) . \end{matrix}$ (▵)

By symmetric role of X₁ and X₂, we only need to prove the following cases and subcases.

(1) a ∈ X₁ and b, c, d ∈ X₂.

(1i) p ∈ X₁ ∖ X₂. Then (▵) is trivial by (MA1).

(1ii) p ∈ X₂.

(1iia) v ∈ X₂. Consider p₁ (a) , b, c, d, p, v ∈ X₂. By (MA3) and M-fuzzifying sand-glass property of $I_{2}$ ,

$\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ = I_{2} (p_{1} (a), d) (p) \land I_{2} (b, c) (p) \\ \land I_{2} (p_{1} (a), b) (v) \\ \leq ⋁_{w \in X} I (c, d) (w) \land I (v, w) (p) . \end{matrix}$

(1iib) v ∈ X₁ ∖ X₂. By (MGI2) and (MGI3), $I (a, b) (v) \leq I_{1} (p_{1} (a), p_{2} (b)) (p_{1} (v)) \leq I_{2} (p_{1} (a), b) (p_{1} (v))$ . Applying (1iia) to p₁ (a) , b, c, d, p, p₁ (v) ∈ X₂, we find that (▵) holds.

(2) a, c ∈ X₁ and b, d ∈ X₂.

(2i) p ∈ X₁.

(2ia) v ∈ X₁.

Consider a, p₂ (b) , c, p₂ (d) , p, v ∈ X₁. By M-fuzzifying sand-glass property of $I_{1}$ ,

$\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ = I_{1} (a, p_{2} (d)) (p) \land I_{1} (c, p_{2} (b)) (p) \\ \land I_{1} (a, p_{2} (b)) (v) \\ \leq ⋁_{w \in X} I (c, d) (w) \land I (v, w) (p) . \end{matrix}$

(2ib) v ∈ X₂. Thus, by (MGI2) and Proposition 3.9, we have $I_{2} (p_{1} (a), b) (v) \leq I_{2} (p_{1} (a), b) (p_{2} (v)) = I_{2} (p_{1} (a), p_{2} (b)) (p_{2} (v)) = I_{1} (p_{1} (a), p_{2} (b)) (p_{2} (v)) = I_{1} (a, p_{2} (b)) (p_{2} (v)) \leq I_{1} (a, p_{2} (b)) (p_{1} (a))$ .

Consider a, p₂ (b) , c, p₂ (d) , p, p₂ (v) ∈ X₁. By (MA3) and M-fuzzifying sand-glass property of $I_{1}$ , we have $\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ = I_{1} (a, p_{2} (d)) (p) \land I_{1} (c, p_{2} (b)) (p) \\ \land I_{1} (a, p_{2} (b)) (p_{2} (v)) \\ \leq ⋁_{w \in X_{1}} I_{1} (c, p_{2} (d)) (w) \land I_{1} (w, p_{2} (v)) (p) \\ \leq ⋁_{w \in X} I (c, d) (w) \land I (v, w) (p) . \end{matrix}$

(2ii) p ∈ X₂. Exchange the labels 1 and 2 in (2i).

(3) a, b ∈ X₁ and c, d ∈ X₂.

(3i) p ∈ X₁.

(3ia) v ∈ X₂ ∖ X₁. Then (▵) holds trivially.

(3ib) v ∈ X₁. If w ∈ X₁ ∖ Y, then $I_{2} (p_{2} (c), p_{2} (d)) (w) = ⊥$ . Consider a, p₂ (c) , p₂ (d) , p, v ∈ X₁. By $I_{2} (p_{2} (a), p_{2} (b)) (w) \leq (p_{2})_{M}^{\to} (I_{2} (a, b)) (w)$ for all a, b ∈ X₂ and w ∈ Y, we have $\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ = I_{1} (a, p_{2} (d)) (p) \land I_{1} (b, p_{2} (c)) (p) \\ \land I_{1} (a, b) (v) \\ \leq ⋁_{w \in X_{1}} I_{1} (p_{2} (c), p_{2} (d)) (w) \land I_{1} (v, w) (p) \\ = ⋁_{w \in Y} I_{1} (p_{2} (c), p_{2} (d)) (w) \land I_{1} (v, w) (p) \\ \leq ⋁_{w \in Y} I_{2} (p_{2} (c), p_{2} (d)) (w) \land I_{1} (v, w) (p) \\ \leq ⋁_{w \in Y} ⋁_{z \in p_{2}^{- 1} (w)} I_{2} (c, d) (z) \land I_{1} (v, p_{2} (z)) (p) \\ = ⋁_{z \in X_{2}} I (c, d) (z) \land I (v, z) (p) \\ \leq ⋁_{z \in X} I (c, d) (z) \land I (v, z) (p) . \end{matrix}$

(3ii) p ∈ X₂. Exchange the labels 1 and 2 in (3i).

(4) a, b, c ∈ X₁ and d ∈ X₂.

(4i) p ∈ X₂ ∖ X₁, or v ∈ X₂ ∖ X₁. Then (▵) is trivial.

(4ii) p, v ∈ X₁. Since a, b, c, p₂ (d) , p, v ∈ X₁, $\begin{matrix} I (a, d) (p) \land I (b, c) (p) \land I (a, b) (v) \\ = I_{1} (a, p_{2} (d)) (p) \land I_{1} (b, c) (p) \land I_{1} (a, b) (v) \\ \leq ⋁_{w \in X_{1}} I_{1} (c, p_{2} (d)) (w) \land I_{1} (v, w) (p) \\ \leq ⋁_{w \in X} I (c, d) (w) \land I (v, w) (p) . \end{matrix}$

(5) a, b, c, d ∈ X₁. Then (▵) follows from $I_{1}$ .

Corollary 4.6. Let $(X_{i}, I_{i})$ (i = 1, 2) be M-fuzzifying geometric interval spaces fulfilling the M-fuzzifying amalgamation condition. If M-fuzzifying gate maps p_i : X_i → X₁ ∩ X₂ are II-mappings, then their M-fuzzifying amalgamation space has sand-glass property iff $I_{i}$ have M-fuzzifying sand-glass property.

Theorem 4.7. Let $(X_{i}, I_{X_{i}})$ and $(Y_{i}, I_{Y_{i}})$ (i = 1, 2) be two pairs of M-fuzzifying geometric interval spaces fulfilling the M-fuzzifying amalgamation condition. Their M-fuzzifying amalgamations are $I_{X}$ and $I_{Y}$ , where X = X₁ ∪ X₂ and Y = Y₁ ∪ Y₂. If f_i : X_i → Y_i (i = 1, 2) are M-fuzzifying IP-mappings coinciding in X₁ ∩ X₂ (X₁∩ X₂ ≠ ∅), then the M-fuzzifying mapping f : X → Y, combined by f_i, is an M-fuzzifying IP-mapping iff f₁ (X₁) ∩ f₂ (X₂) = f (X₁ ∩ X₂).

Proof. Necessity. If y ∈ f₁ (X₁) ∩ f₂ (X₂), then there exists x_i ∈ X_i such that y = f_i (x_i). Since $f_{M}^{\to}$ is an M-fuzzifying IP-mapping, $y_{⊤} \leq f_{M}^{\to} (I_{X} (x_{1}, x_{2})) \leq I_{Y} (f (x_{1}), f (x_{2})) = y_{⊤}$ . Also, $f_{M}^{\to} (I_{X} (x_{1}, x_{2})) = f_{M}^{\to} (I_{1} (x_{1}, p_{X_{2}} (x_{2})) \lor I_{1} (p_{X_{1}} (x_{1}), x_{2}))$ . Thus $⊤ = f_{M}^{\to} (I_{X} (x_{1}, x_{2})) (f (p_{X_{1}} (x_{1}))) = y_{⊤} (f (p_{X_{1}} (x_{1})))$ . So f (p_{X
₁} (x₁)) = y. Since p_{X
₁} (x₁) ∈ X₁ ∩ X₂, y ∈ f (X₁ ∩ X₂). Thus f₁ (X₁) ∩ f₂ (X₂) = f (X₁ ∩ X₂).

Sufficiency. Let a, b ∈ X. If a, b ∈ X_i, then $f_{M}^{\to} (I_{X} (a, b)) \leq I_{Y} (f (a), f (b))$ . If a ∈ X₁ and b ∈ X₂, then by Theorem 3.10 and (MA3), we have

$\begin{matrix} f_{M}^{\to} (I_{X} (a, b)) \\ = (f_{1})_{M}^{\to} (I_{X_{1}} (a, p_{X_{2}} (b))) \lor (f_{2})_{M}^{\to} (I_{X_{2}} (p_{X_{1}} (a), b)) \\ \leq I_{Y_{1}} (f_{1} (a), f_{2} (p_{X_{2}} (b))) \lor I_{Y_{2}} (f_{1} (p_{X_{1}} (a)), f_{2} (b)) \\ \leq I_{Y_{1}} (f_{1} (a), p_{Y_{2}} (f_{2} (b))) \lor I_{Y_{2}} (p_{Y_{1}} (f_{1} (a)), f_{2} (b)) \\ = I_{Y} (f (a), f (b)) . \end{matrix}$

Therefore $f_{M}^{\to}$ is an M-fuzzifying IP-mapping.

Theorem 4.8. Let $(X_{i}, I_{i})$ be disjoint M-fuzzifying geometric interval spaces with M-fuzzifying gated sets A_i ⊆ X_i (i = 1, 2). If f : A₁ → A₂ is an M-fuzzifying II-isomorphism with respect to M-fuzzifying geometric interval spaces $(A_{i}, I_{i} ∣_{A_{i}})$ (i = 1, 2), where $I_{i} ∣_{A_{i}}$ is the restriction of $I_{i}$ on A_i, then there exists an unique M-fuzzifying geometric interval operator $I$ on X₁ ∪ X₂ = X such that

$I$ extends both $I_{1}$ and $I_{2}$ ;

X_i (i = 1, 2) are gated sets with respect to $(X, I)$ . In addition, the M-fuzzifying gate of a_i in X_j is coincide with the M-fuzzifying gate of a_i in A_j for each a_i ∈ X_i (i, j = 1, 2, i ≠ j).

f is the M-fuzzifying mutual gate map of A₁ and A₂. That is, for each a ∈ A₁, a and f (a) are mutual gets in A₁ and A₂.

Proof. Let $I : X \times X \to M^{X}$ be defined by:

(A) for all a, b ∈ X_i (i = 1, 2), $\begin{matrix} I (a, b) (z) = {\begin{matrix} I_{i} (a, b) (z), & z \in X_{i}, \\ ⊥, & z \notin X_{i}; \end{matrix} \end{matrix}$

(B) for all a ∈ X₁ and b ∈ X₂, $\begin{matrix} I (a, b) (z) & = & {\begin{matrix} I_{1} (a, f^{- 1} (p_{2} (b)) (z), & z \in X_{1}, \\ I_{2} (f (p_{1} (a)), b) (z), & z \in X_{2} . \end{matrix} \end{matrix}$

We prove the results by the following steps.

Step 1. $I$ is M-fuzzifying geometric.

Clearly, $I$ satisfies (MI1), (MI2), (MGI1) and (i). To prove (MGI2&3), we prove for all a, b, u, v ∈ X, $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ \leq I (a, b) (u) \land I (u, b) (v) . \end{matrix}$ (∇)

We prove (∇) by the following four cases.

(1) a, b ∈ X₁. If v ∉ X₁ or u ∉ X₁, then (∇) is trivial. If v, u ∈ X₁, then (∇) follows from $I_{1}$ .

(2) a ∈ X₁ and b ∈ X₂.

(2a) u, v ∈ X₁. By (MGI2&3) of $I_{1}$ , we have $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ = I_{1} (a, f^{- 1} (p_{2} (b))) (v) \land I_{1} (a, v) (u) \\ \leq I_{1} (a, f^{- 1} (p_{2} (b))) (u) \land I_{1} (u, f^{- 1} (p_{2} (b))) (v) \\ = I (a, b) (u) \land I (u, b) (v) . \end{matrix}$

(2b) v ∈ X₁ and u ∈ X₂. Then (Δ) is trivial.

(2c) v ∈ X₂ and u ∈ X₁. By (MGI2&3) of $I_{1}$ , Proposition 3.9 and (MGI2) of $I_{2}$ , we have $\begin{matrix} I_{1} (a, f^{- 1} (p_{2} (v)) (u) \\ \leq I_{1} (a, f^{- 1} (p_{2} (v)) (p_{1} (u)) \land I_{1} (a, p_{1} (u)) (u) \\ = I_{2} (f (p_{1} (a)), p_{2} (v)) (f (p_{1} (u))) \land I_{1} (a, p_{1} (u)) (u) \\ \leq I_{2} (f (p_{1} (a)), v) (f (p_{1} (u))) \land I_{1} (a, p_{1} (u)) (u) . \end{matrix}$

Thus by (MGI2&3) of $I_{2}$ , Proposition 3.9, Theorem 3.10, (MGI2) of $I_{1}$ and the M-fuzzifying II-isomorphism f, we have $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ = I_{2} (f (p_{1} (a)), b) (v) \land I_{1} (a, f^{- 1} (p_{2} (v))) (u) \\ \leq I_{1} (a, p_{1} (u)) (u) \land {I_{2} (f (p_{1} (a)), b) (v) \\ \land I_{2} (f (p_{1} (a)), v) (f (p_{1} (u)))} \\ \leq I_{1} (a, p_{1} (u)) (u) \land {I_{2} (f (p_{1} (a)), b) (f (p_{1} (u))) \\ \land I_{2} (f (p_{1} (u)), b) (v)} \\ = I_{1} (a, p_{1} (u)) (u) \land I_{1} (p_{1} (a), \\ f^{- 1} (p_{2} (b))) (p_{1} (u)) \land I (u), b) (v) \\ \leq I_{1} (a, f^{- 1} (p_{2} (v))) (u) \land I (u, b) (v) \\ = I (a, b) (u) \land I (u, b) (v) . \end{matrix}$

(2d) u, v ∈ X₂. By (MGI2&3) of $I_{2}$ , we have $\begin{matrix} I (a, b) (v) \land I (a, v) (u) \\ = I_{2} (f (p_{1} (a)), b) (v) \land I_{2} (f (p_{1} (a)), v) (u) \\ \leq I_{2} (f (p_{1} (a)), b) (u) \land I_{2} (u, b) (v) \\ = I (a, b) (u) \land I (u, b) (v) . \end{matrix}$

(3) a ∈ X₂ and b ∈ X₁.

Exchange the labels 1 and 2 in (2), we obtain (∇).

(4) a, b ∈ X₂. The result is similar to (1).

Step 2. To prove (ii), let a ∈ X₁ and b ∈ X₂. Then $I (a, b) (f^{- 1} (p_{2} (b))) = ⊤$ . Thus f^-1 (p₂ (b)) is the M-fuzzifying gate of b in X₁ with respect to $(X, I)$ . Similarly, $I (b, a) (f (p_{1} (a))) = ⊤$ . Hence f (p₁ (a)) is the M-fuzzifying gate of a in X₂ with respect to $(X, I)$ . So X_i (i = 1, 2) are gated sets with respective M-fuzzifying gate maps P_i : X → X_i satisfying $\begin{matrix} P_{1} (b) & = & {\begin{matrix} b, & b \in X_{1}, \\ f^{- 1} (p_{2} (b)), & b \in X_{2}; \end{matrix} \\ P_{2} (a) & = & {\begin{matrix} a, & a \in X_{2}, \\ f (p_{1} (a)), & a \in X_{1} . \end{matrix} \end{matrix}$

Clearly, for a ∈ X_i, P_i (a) ⊢ _{A
_j}a (i, j = 1, 2, i ≠ j).

Step 3. To prove (iii), let c ∈ A₁.

For each x ∈ A₂, we have $I (x, c) (f (c)) = I_{2} (x, f (p_{1} (c))) (f (c)) = I_{2} (x, f (c)) (f (c)) = ⊤$ . Thus f (c) is the M-fuzzifying gate of c in A₂. Similarly, for each y ∈ A₁, we have $I (y, f (c)) (c) = I_{1} (y, f^{- 1} (p_{2} (f (c)))) (c) = I_{1} (x, c) (c) = ⊤$ . Hence c is the M-fuzzifying gate of f (c) in A₁.

Therefore c and f (c) are mutual gates in A₁ and A₂.

Step 4. let $\bar{I}$ be an M-fuzzifying geometric interval operator on X satisfying (i), (ii) and (iii).

To prove $\bar{I} = I$ , let a ∈ X₁, b ∈ X₂ and z ∈ X.

If z ∈ X₁, then $\bar{I} (a, b) (f^{- 1} (p_{2} (b))) = ⊤$ by (ii). Thus by (i) and (MGI2) of $\bar{I}$ , we have $I (a, b) (z) = I_{1} (a, f^{- 1} (p_{2} (b))) (z) = \bar{I} (a, f^{- 1} (p_{2} (b))) (z) \leq \bar{I} (a, b) (z)$ . Conversely, $\bar{I} (z, b) (f^{- 1} (p_{2} (b))) = ⊤$ by (ii). Thus, by (MGI2&3) and (i), $\bar{I} (a, b) (z) \leq \bar{I} (a, f^{- 1} (p_{2} (b))) (z) = I_{1} (a, f^{- 1} (p_{2} (b))) (z) = I (a, b) (z)$ . Hence $\bar{I} (a, b) (z) = I (a, b) (z)$ .

If z ∈ X₂, then $\bar{I} (a, b) (f (p_{1} (a)) = ⊤$ by (ii). So $I (a, b) (z) = I_{2} (f (p_{1} (a)), b) (z) = \bar{I} (f (p_{1} (a)), b) (z) \leq \bar{I} (a, b) (z)$ by (i) and (MGI2) of $\bar{I}$ . Conversely, since $\bar{I} (u, b) (f^{- 1} (p_{2} (b))) = ⊤$ by (ii), we have $\bar{I} (a, b) (z) \leq I (f (p_{1} (a)), b) (z) = I_{2} (f (p_{1} (a)), b) (z) = I (a, b) (z)$ by (MGI2&3) of $\bar{I}$ and (i). Hence $\bar{I} (a, b) (z) = I (a, b) (z)$ . Therefore $\bar{I} = I$ .

5 Conclusions

M-fuzzifying base-point orders constructed by M-fuzzifying interval spaces are M-fuzzifying quasi-orders which can be used to characterize M-fuzzifying geometric (resp. Peano, Pasch) property (see Theorem 3.3 and 3.5). They can also be used as an alternative method to define M-fuzzifying gates (see Proposition 3.7(3)). M-fuzzifying gated maps derived from M-fuzzifying gates are necessary in constructing M-fuzzifying amalgamations of M-fuzzifying geometric interval spaces. The method used in Theorem 4.1 not only provide an natural way of constructing M-fuzzifying geometric intervals from small pieces, but also shows that the newly constructed M-fuzzifying geometric interval operators preserve many properties that preserved by the two original M-fuzzifying interval spaces (see Theorems 4.3, 4.5, and Corollarys 4.2, 4.4 and 4.6).

Footnotes

Acknowledgments

The authors sincerely appreciate the editor for his great help, and the reviewers’ valuable suggestions.

This work is supported by the National Natural Science Foundation of China (No. 11471202) and the Educational Commission Foundation of Hunan province (No. 15C0586).

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