**1. 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 [9–13]. 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 [16–21]. 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

**Citation:** Dong, Y.-Y.; Shi, F.-G. *L*-Fuzzy Sub-Effect Algebras. *Mathematics* **2021**, *9*, 1596. https:// doi.org/10.3390/math9141596

Academic Editor: Sorin Nadaban

Received: 13 May 2021 Accepted: 29 June 2021 Published: 7 July 2021

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about *M*-fuzzifying convex structures showed in [23–26]. In 2017, abstract convexity was extended to a more general case, which is called an (*<sup>L</sup>*, *M*)-fuzzy convex structure in [27]. Particularly, an (*<sup>L</sup>*, *L*)-fuzzy structure is briefly called an *L*-fuzzy convex structure. Many researchers investigated (*<sup>L</sup>*, *M*)-fuzzy convex structures from different aspects [28–32]. 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 *L*subalgebras on effect algebras. In Section 4, we obtain an *L*-fuzzy convexity induced by *L*fuzzy 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.
