L -Fuzzy Sub-Effect Algebras

: In this paper, the notions of L -fuzzy subalgebra degree and L -subalgebras on an effect algebra are introduced and some characterizations are given. We use four kinds of cut sets of L -subsets to characterize the L -fuzzy subalgebra degree. We induce an L -fuzzy convexity by the L -fuzzy subalgebra degree, and we prove that a morphism between two effect algebras is an L -fuzzy convexity preserving mapping and a monomorphism is an L -fuzzy convex-to-convex mapping. Finally, it is proved that the set of all L -subalgebras on an effect algebra can form an L -convexity, and its L -convex hull


Introduction
Effect algebras were introduced by Foulis and Bennett to axiomatize quantum logic effects on a Hilbert space [1]. The elements of effect algebras represent events that may be unsharp or imprecise. Effect algebras are partial algebras with one partial binary operation that can be converted into bounded posets in general and into lattices in some cases. Since Zadeh introduced the concept of fuzzy sets [2], the theory of fuzzy sets has become a vigorous area of research in different disciplines. In recent years, many scholars have studied (fuzzy) ideals [3,4] and (fuzzy) filters [5,6] (the lattice background is [0, 1]) on effect algebras, but there is no research on the fuzzy subalgebras (L-subalgebras). In order to fill this gap, we first want to extend the unit interval [0, 1] to a completely distributive lattice L and introduce the notion of L-subalgebras on effect algebras. As we all know, for a given L-subset A, A is either an L-subalgebra or not, so let us consider the question: what is the degree to which A is an L-subalgebra? To solve this question, we propose the concept of L-subalgebra degree on effect algebras and investigate their basic properties in this paper.
The notion of convexity [7] is inspired by the shape of some figures, such as circles and polyhedrons in Euclidean spaces. As we all know, a convex structure satisfying the Exchange Law [8] is a matroid, matroids are precisely the structures for which the very simple and efficient greedy algorithm works. Many real-world problems can be defined and solved by making use of matroid theory. So convexity theory has been regarded as an increasingly important role in solving problems. In fact, there exist convexities in many mathematical structures such as semigroup, ring, posets, graphs, convergence spaces and so on [9][10][11][12][13]. It is also natural to consider if there exist convex structures on effect algebras. By these motivations, we will try to prove the existence of convexity on effect algebras.
With the development of fuzzy sets, the notion of convexities has already been extended to fuzzy case. In 1994, Rosa [14] first proposed the notion of fuzzy convex structures with the unit interval [0, 1] as the lattice background. In 2009, Y. Maruyama [15] defined another more generalized fuzzy convex structure based on a completely distributive lattice L, which is called L-convex structure, some of the latest research related to L-convex structure can be found in [16][17][18][19][20][21]. In 2014, a new approach to the fuzzification of convex structures was introduced in [22]. It is called an M-fuzzifying convex structure, in which each subset can be regarded as a convex set to some degrees. Further, there are some studies about M-fuzzifying convex structures showed in [23][24][25][26]. In 2017, abstract convexity was extended to a more general case, which is called an (L, M)-fuzzy convex structure in [27]. Particularly, an (L, L)-fuzzy structure is briefly called an L-fuzzy convex structure. Many researchers investigated (L, M)-fuzzy convex structures from different aspects [28][29][30][31][32]. In our paper, we mainly discuss L-fuzzy convex structure and L-convex structure on an effect algebra.
The paper is organized in the following way. In Section 2, we will give some necessary notations and definitions. In Section 3, we propose the notion of L-fuzzy subalgebra degree on an effect algebra by means of the implication operator of L, and we provide their characterizations by cut sets of L-subsets. For instance, we give the notion of Lsubalgebras on effect algebras. In Section 4, we obtain an L-fuzzy convexity induced by Lfuzzy subalgebra degree, and we analyze the corresponding L-fuzzy convexity preserving mappings and L-fuzzy convex-to-convex mappings. Finally, we prove that the set of all L-subalgebras on an effect algebra can form an L-convexity, and we provide its L-convex hull formula.

Preliminaries
In this section, we provide some notions and results that will be used in this paper.

Definition 1 ([1]
). An effect algebra is a system (E, ⊕, 0, 1) consisting of a set E with two special elements 0, 1 ∈ E, called the zero and the unit, and with a partially defined binary operation ⊕ satisfying the following conditions: for any x, y, z ∈ E, E3) (Orthosupplement law) for every x ∈ E there exists a unique x ∈ E such that x ⊕ x is defined and x ⊕ x = 1. The unique element x ∈ E called the orthosupplement of x, (E4) (Zero-one law) if x ⊕ 1 is defined, then x = 0.
In the following, an effect algebra (E, ⊕, 0, 1) is denoted by E unless otherwise specified.

Definition 2 ( [33]).
A nonempty subset S of an effect algebra E is called a subalgebra if it satisfies the following conditions: Let E be an effect algebra. If x ⊕ y is defined, then we say x ⊥ y for all x, y ∈ E. Define a binary relation on E by x ≤ y if for some z ∈ E, x ⊕ z = y, which turns out to be a partial ordering on E such that 0 and 1 is the smallest element and the greatest element of E, respectively. If the poset (E, ≤) is a lattice, then E is called a lattice-ordered effect algebra.
Let L be a complete lattice. We denote the minimal element and the maximal element of L by ⊥ L and L , respectively. An element λ ∈ L is called co-prime if λ ≤ δ ∨ ξ implies λ ≤ δ or λ ≤ ξ. The set of nonzero co-prime elements in L is denoted by J(L). An element λ ∈ L is called prime if λ ≥ δ ∧ ξ implies λ ≥ δ or λ ≥ ξ. The set of nonunit prime elements in L is denoted by P(L). From [34], we know that each element of L is the sup of co-prime elements and the inf of prime elements.
In the following, a completely distributive lattice (L, ∧, ∨) is denoted by L unless otherwise specified.
For an effect algebra E, each mapping A : E −→ L is called an L-subset of E, and we denote the collection of all L-subsets of E by L E . L E is also a complete lattice by defining "≤" point-wisely. Furthermore, the smallest element and the largest element in L E are denoted by ⊥ and , respectively. The mapping g → L : ).

Definition 3 ([36])
. Let E be an effect algebra and A ∈ L E . For any λ ∈ L, define The right adjoint → of the meet operation ∧ is a mapping from L × L to L defined as Some basic properties of the operation → are listed in the following [37,38].
Let ⊥, ∈ L E represent ⊥(x) = ⊥ L and (x) = L for all x ∈ E. Next, we recall (L, M)-fuzzy convexities in [27], which are more general fuzzy convexities. Let L and M be two completely distributive lattices. An (L, M)-fuzzy convexity on an effect algebra E is defined as follows: . A mapping C : L E −→ M is called an (L, M)-fuzzy convexity if it satisfies the following conditions: In this case, the pair (E, C) is called an (L, M)-fuzzy convex space. An (L, L)-fuzzy convexity is briefly called an L-fuzzy convexity.

Definition 5 ([27]
). Let (E, C) and (F, D) be two (L, M)-fuzzy convex spaces. If g : E −→ F is a mapping between E and F, then is called an (L, M)-fuzzy convex-to-convex mapping provided that D(g → L (B)) ≥ C(B) for all B ∈ L E . An (L, L)-fuzzy convexity preserving mapping and an (L, L)-fuzzy convex-to-convex mapping are briefly called an L-fuzzy convexity preserving mapping and an L-fuzzy convex-to-convex mapping, respectively.

L-Fuzzy Subalgebra Degree on Effect Algebras
In this section, we introduce the concept of L-fuzzy subalgebra degree on effect algebras by means of the implication operator of L. We define an L-fuzzy subalgebra provided that its L-fuzzy subalgebra degree is equal to L . Moreover, we give some characterizations of L-subalgebra degree in terms of four kinds of cut sets of L-subsets. Definition 8. Let E be an effect algebra and A ∈ L E . Then the L-fuzzy subalgebra degree E (A) of A is defined as: Example 1. Let E = {0, x, y, 1} and ⊕ be given by: is an effect algebra. Let L = [0, 1] and define the L-subsets of E as follows: We can easily know x = x, y = y, 0 = 1 and From the properties (3) of the implication operator, the following lemma is obvious.

Lemma 1.
Let E be an effect algebra and A ∈ L E . Then E (A) ≥ λ (λ ∈ L) if and only if for any x, y ∈ E satisfying x ⊥ y, By Lemma 1, we can obtain the following theorem.
Theorem 1. Let E be an effect algebra and A ∈ L E . Then We use four kinds of cut sets of L-subsets to characterize the L-fuzzy subalgebra degree in the following Theorem.

Theorem 2. Let E be an effect algebra and A
Proof. (i) For any x, y ∈ E satisfying x ⊥ y, assume that λ ∈ L with the property of Conversely, assume λ ∈ L and for each µ ≤ λ, (ii) For any x, y ∈ E satisfying x ⊥ y, assume that λ ∈ L with the property of . For any µ ∈ P(L) with µ λ, suppose A (µ) = ∅ and let x, y ∈ A (µ) with x ⊥ y, i.e., A(x) µ and A(y) µ, then Conversely, assume that λ ∈ L and A (µ) = ∅ or it is a subalgebra of E for µ ∈ P(L) with µ λ. For any x, y ∈ E satisfying x ⊥ y, let µ ∈ P(L) and A(x) ∧ A(y) ∧ λ µ, then x, y ∈ A (µ) and λ µ.
(iii) For any x, y ∈ E satisfying x ⊥ y, assume that λ ∈ L with the property of Conversely, assume that λ ∈ L and A [µ] = ∅ or it is a subalgebra of E for µ / ∈ α(λ).
Similarly, we can prove (iv) By (iii) we first obtain Conversely, assume that λ ∈ L and A [µ] = ∅ or it is a subalgebra of E for all µ ∈ P(L) .

Similarly, we can prove
This completes the proof. Definition 9. Let E be an effect algebra and A ∈ L E . Then A is called an L-subalgebra provided that E (A) = L . In particular, we say an L-subalgebra is a fuzzy subalgebra when L = [0, 1].

Example 2.
In Example 1, it follows from Definition 9 that A 1 is a fuzzy subalgebra, but A 2 and A 3 are not fuzzy subalgebras; A 2 is a fuzzy subalgebra in the degree of 0.3.
From Definition 9 and Lemma 1, the following proposition is obvious.

Proposition 1. Let E be an effect algebra and A ∈ L E . Then A is an L-subalgebra if and only if
for any x, y ∈ E with x ⊥ y,

Proof. (i) Since x ⊥ x and A is an L-subalgebra of E, we have
(ii) By Proposition 1 we first obtain A(x ) ≥ A(x). On the other hand, by Definition 1 (E3), we know x = x and thus A(x) = A(x ) ≥ A(x ).
By Definition 9 and Theorem 2, we can obtain the following two results Corollary 2. Let A be an L-subset of an effect algebra E. Then the following conditions are equivalent: (v) for each λ ∈ P(L), A [λ] = ∅ or it is a subalgebra of E; (vi) for each λ ∈ P(L), A (λ) = ∅ or it is a subalgebra of E.
Let E be an effect algebras and (2 E ) L represents the set of all mapping H : L −→ 2 E . Then we have the following definitions.
In particular, when L = [0, 1], we have the following result.

L-Fuzzy Convexity on Effect Algebras
In this section, we study the relationship between the L-fuzzy subalgebra degree and L-fuzzy convexity on an effect algebra. Further, we prove that a morphism between two effect algebras is an L-fuzzy convexity preserving mapping and a monomorphism is an L-fuzzy convex-to-convex mapping. Finally, we prove that the set of all L-subalgebras on an effect algebra is an L-convexity. For instance, we give its L-convex hull formula.
For each A ∈ L E , E can be naturally considered as a mapping E : L E −→ L defined by A −→ E (A). The following theorem will show that E is an L-fuzzy convexity on an effect algebra E.
Theorem 5. Let E be an effect algebra. Then E : L E −→ L is an L-fuzzy convexity on E.

Proof. (LMC1). It is clear that
then there exists i, j ∈ I such that µ ≤ A i (x), µ ≤ A j (y) and µ ≤ λ.
Similarly, we can prove Therefore, E : L E −→ L is an L-fuzzy convexity.
In order to investigate L-fuzzy convexity preserving mappings and L-fuzzy convexto-convex mappings, we first give following definition and lemma. ([40]). Let E and F be effect algebras and g: E −→ F is called (i) a morphism if g(1 E ) = 1 F , and x ⊥ y, x, y ∈ E implies g(x) ⊥ g(y) in F, and g(x ⊕ y) = g(x) ⊕ g(y); (ii) a monomorphism if g is a morphism and g(x) ⊥ g(y) iff x ⊥ y.

Lemma 2.
Let E and F be effect algebras and g: E −→ F is a morphism. Then g(0 E ) = 0 F and Theorem 6. Let g : E −→ F be a morphism between two effect algebras E and F. Then Proof. (i) In order to prove g: (E, E E ) −→ (M, E F ) is L-fuzzy convexity preserving, we just need to prove for any Similarly, we can obtain λ ∧ g ← It follows that for x, y ∈ F satisfying x ⊥ y, Similarly, we can prove (g → Proof. It is easy to check that p i : ∏ i∈Λ E i −→ E i is a morphism between effects algebras, so by Theorem 6, we know p i is L-fuzzy convexity preserving for all i ∈ Λ. Theorem 7. Let {E i } i∈Λ be a family of effect algebras and ∏ i∈Λ A i be the direct product of {A i } i∈Λ , So by the proof of (LMC2) in Theorem 8, we have The last inequality holds because p i is L-fuzzy convexity preserving by Corollary 4.
From paper [27], we know (E, E λ ) is an L-convex space for all λ ∈ L\⊥ L . In particular, (E, E L ) is an L-convex space. We should note that E L = {A ∈ L E | E (A) = L }, by Definition 9, we obtain that E L is the set of all L-subalgebras on an effect algebra E.
The L-convex hull operator with respect to the L-convex space (E, E L ) is as follows: [17], i.e., co(A) is the least L-subalgebra containing A. In the following, we will give the corresponding L-convex hull formula. Theorem 8. Let A be an L-subset of an effect algebra E. Define for any x ∈ E, Then co(A) = ∞ n=0 σ n .
Proof. We first show ∞ n=0 σ n is an L-subalgebra of E. To achieve this, we give the following statements.
(i). For any x ∈ E and n ≥ 1, σ n (1) ≥ σ n (x). Let n = 1. For any x ∈ E, it follows from x ⊕ x = 1 that Let n ≥ 2. For any x ∈ E, since x ⊕ x = 1, we have that for each x ∈ E.

Conclusions
In this paper, we proposed the notion of L-fuzzy subalgebra degree on an effect algebra based on a completely distributive lattice L. We gave their characterizations in terms of four kinds of cut sets of L-subsets. We say an L-subset is an L-subalgebra if its L-fuzzy subalgebra degree is equal to L . In fact, the L-fuzzy subalgebra degree can describe the degree to which an L-subset is an L-fuzzy subalgebra. An L-fuzzy convex structure on an effect algebra was naturally constructed and some of its properties were studied. For instance, an L-convex structure was induced by the set of all L-subalgebras and its L-convex hull formula was given.
It should be noted that the same thought can be applied to different algebraic structure such as MV-algebras, BL-algebras, residuated lattices and so on. Thus, there exist L-fuzzy convexities in many logical algebras, which enrich the convexity theory in logical algebras.

Acknowledgments:
The authors express thanks to the handling Editor and the reviewers for their careful reading and useful suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.