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
) 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
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
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
-fuzzy convex structure in [
27]. Particularly, an
-fuzzy structure is briefly called an
L-fuzzy convex structure. Many researchers investigated
-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
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.
2. 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 consisting of a set E with two special elements , called the zero and the unit, and with a partially defined binary operation ⊕ satisfying the following conditions: for any ,- (E1)
(Commutative law) if is defined, then is defined and ,
- (E2)
(Associative law) if is defined and is defined, then and are defined and ,
- (E3)
(Orthosupplement law) for every there exists a unique such that is defined and . The unique element called the orthosupplement of x,
- (E4)
(Zero-one law) if is defined, then .
In the following, an effect algebra 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:- (i)
,
- (ii)
implies ,
- (iii)
if and is defined, then .
Let E be an effect algebra. If is defined, then we say for all . Define a binary relation on E by if for some , , 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 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
and
, respectively. An element
is called co-prime if
implies
or
. The set of nonzero co-prime elements in
L is denoted by
. An element
is called prime if
implies
or
. The set of nonunit prime elements in
L is denoted by
. From [
34], we know that each element of
L is the sup of co-prime elements and the inf of prime elements.
Let
, the symbol
(
is wedge below
) means that for every
,
implies the existence of
such that
. A complete lattice
L is completely distributive [
34] if and only if
for each
. The set
, denoted by
, is called the greatest minimal family of
in the sense of [
34]. Let
. Moreover, define a binary relation
as follows: for
,
if and only if for every subset
,
implies
for some
. The set
, denoted by
, is the greatest maximal family of
in the sense of [
34]. Let
. We know that
is an
mapping and
is a union-preserving mapping, it holds that
for each
(see [
34]). From [
35], we have
and
.
In the following, a completely distributive lattice is denoted by L unless otherwise specified.
For an effect algebra
E, each mapping
is called an
L-subset of
E, and we denote the collection of all
L-subsets of
E by
.
is also a complete lattice by defining “≤” point-wisely. Furthermore, the smallest element and the largest element in
are denoted by
and
, respectively. The mapping
is induced by
as follows:
is induced by the mapping
g as follows:
.
Definition 3 ([
36]).
Let E be an effect algebra and . For any , define
The right adjoint → of the meet operation ∧ is a mapping from
to
L defined as
Some basic properties of the operation → are listed in the following [
37,
38].
- (1)
;
- (2)
;
- (3)
and ;
- (4)
.
Let
represent
and
for all
. Next, we recall
-fuzzy convexities in [
27], which are more general fuzzy convexities. Let
L and
M be two completely distributive lattices. An
-fuzzy convexity on an effect algebra
E is defined as follows:
Definition 4 ([
27]).
A mapping is called an -fuzzy convexity if it satisfies the following conditions:- (LMC1)
;
- (LMC2)
if is nonempty, then ;
- (LMC3)
if is nonempty and directed, then .
In this case, the pair is called an -fuzzy convex space. An -fuzzy convexity is briefly called an L-fuzzy convexity.
An
-fuzzy convexity is an
L-convexity in [
15]. An
-fuzzy convexity is a fuzzy convexity in [
14], where
. A
-fuzzy convexity is an
M-fuzzifying convexity in [
22]. A
-fuzzy convexity is a convexity in [
7].
Definition 5 ([
27]).
Let and be two -fuzzy convex spaces. If is a mapping between E and F, then- (i)
is called an -fuzzy convexity preserving mapping provided that for all .
- (ii)
is called an -fuzzy convex-to-convex mapping provided that for all .
An -fuzzy convexity preserving mapping and an -fuzzy convex-to-convex mapping are briefly called an L-fuzzy convexity preserving mapping and an L-fuzzy convex-to-convex mapping, respectively.
An
-fuzzy convexity preserving mapping is an
L-convexity preserving mapping in [
15]. An
-fuzzy convex-to-convex mapping is an
L-convex-to-convex mapping in [
15].
Definition 6 ([
1]).
Let be a family of effect algebras, define as:Define the operation ⊕ on as: for , iff for all . In this case, and . Further, where and are the minimal element and the maximal element of . Then is called the direct product of effect algebras.
It is easy to check that is an effect algebra.
Definition 7 ([
39]).
Let be a family of effect algebras and be an L-subset of for all , then the L-subset of is defined by . 3. -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 . 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 . Then the L-fuzzy subalgebra degree of A is defined as: Example 1. Let and ⊕ be given by: ⊕ | 0 | x | y | 1 |
0 | 0 | x | y | 1 | |
x | x | 1 | ∗ | ∗ | |
y | y | ∗ | 1 | ∗ | |
1 | 1 | ∗ | ∗ | ∗ | |
Then is an effect algebra. Let and define the L-subsets of E as follows:
- (i)
. By Definition 8, we can obtain .
Indeed, since for any , , we have for all . We can easily know and , it follows from that , so for all . We can routinely prove for all with . Therefore, .
- (ii)
. By Definition 8, we can obtain .
Indeed, since , we can easily obtain .
- (iii)
. By Definition 8, we can obtain .
Indeed, since , we have .
From the properties (3) of the implication operator, the following lemma is obvious.
Lemma 1. Let E be an effect algebra and . Then () if and only if for any satisfying , By Lemma 1, we can obtain the following theorem.
Theorem 1. Let E be an effect algebra and . 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 . Then
- (i)
- (ii)
- (iii)
- (iv)
- (v)
if for all .
Proof. (i) For any
satisfying
, assume that
with the property of
and
. For any
, suppose
and let
with
, then
which implies
and
. Hence
is a subalgebra of
E. This gives that
Conversely, assume
and for each
,
or it is a subalgebra of
E. For any
satisfying
, let
and
, then
and
. It follows that
and
. Since
and
are subalgebras of
E, we have
and
, i.e.,
and
. So we have
(ii) For any
satisfying
, assume that
with the property of
and
. For any
with
, suppose
and let
with
, i.e.,
and
, then
since
is prime. By
and
, it follows that
and
, i.e.,
. Hence
is a subalgebra of
E. This shows that
Conversely, assume that
and
or it is a subalgebra of
E for
with
. For any
satisfying
, let
and
, then
and
. Since
is a subalgebra of
E, it holds that
, i.e.,
. Hence
. Similarly, we can prove
. This shows
(iii) For any
satisfying
, assume that
with the property of
and
. For any
, suppose
and let
with
, i.e.,
and
, then by
, we have
. From
we know
and
. By
and
, we have
,
and
, i.e.,
. Hence
is a subalgebra of
E. This shows
Conversely, assume that
and
or it is a subalgebra of
E for
. Now we prove
and
for all
satisfying
. Let
and
, then
. Thus
and
. Since
is a subalgebra of
E, it holds that
, i.e.,
. This means
. Hence
Similarly, we can prove
. This shows
(iv) By (iii) we first obtain
Conversely, assume that
and
or it is a subalgebra of
E for all
satisfying
. Now we prove
and
for all
satisfying
. Let
and
. It follows that
Thus
and
. Since
is a subalgebra of
E, it holds that
, i.e.,
. So
, this means
. Hence
Similarly, we can prove
. This shows
(v) For any
satisfying
, assume that
with the property of
and
. For any
, suppose
and let
with
, i.e.,
and
. Then by
, we have
. From
we know
and
. By
and
, we have
,
and
, i.e.,
. Hence
is a subalgebra of
E. This shows
Conversely, assume that
and
or it is a subalgebra of
E for
. Now we prove
and
for all
satisfying
. The statement holds obviously when
. Assume
, it implies
and
. Take
with
, then
. Thus
and
. Since
is a subalgebra of
E, it holds that
, i.e.,
. This means
. Hence
Similarly, we can prove
. This shows
This completes the proof. □
Definition 9. Let E be an effect algebra and . Then A is called an L-subalgebra provided that . In particular, we say an L-subalgebra is a fuzzy subalgebra when .
Example 2. In Example 1, it follows from Definition 9 that is a fuzzy subalgebra, but and are not fuzzy subalgebras; 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 . Then A is an L-subalgebra if and only if for any with , Corollary 1. Let A be an L-subalgebra of an effect algebra E. Then
- (i)
for each , ;
- (ii)
for each , ;
- (iii)
.
Proof. (i) Since
and
A is an
L-subalgebra of
E, we have
(ii) By Proposition 1 we first obtain . On the other hand, by Definition 1 (E3), we know and thus .
(iii) By (i) we know . By Proposition 1, we have , so . □
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:
- (i)
A is an L-subalgebra;
- (ii)
for each , or it is a subalgebra of E;
- (iii)
for each , or it is a subalgebra of E;
- (iv)
for each , or it is a subalgebra of E;
- (v)
for each , or it is a subalgebra of E;
- (vi)
for each , or it is a subalgebra of E.
Corollary 3. Let A be an L-subset of an effect algebra E. If for all , then the following conditions are equivalent:
- (i)
A is an L-subalgebra;
- (ii)
for each , or it is a subalgebra of E;
- (iii)
for each , or it is a subalgebra of E.
Let E be an effect algebras and represents the set of all mapping . Then we have the following definitions.
Definition 10 ([
36]).
Let .- (i)
If implies , then H is called an -nest of E.
- (ii)
If implies , then H is called an -nest of E.
By Theorem 2.3 in [
36], we can prove the following two theorems.
Theorem 3. Let L be completely distributive and let () be an -nest of subalgebras on an effect algebra E. Then there exists an L-subalgebra B such that
- (i)
for all ;
- (ii)
for all ;
- (iii)
for all .
Theorem 4. Let L be completely distributive and let () be an -nest of subalgebras on an effect algebra E. Then there exists an L-subalgebra B such that
- (i)
for all ;
- (ii)
for all ;
- (iii)
for all .
In particular, when , we have the following result.
Corollary 4. Let () be a family of subalgebras on an effect algebra E. If , then there exists a fuzzy subalgebra B satisfying
- (i)
;
- (ii)
for all ;
- (iii)
for all .
4. -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 , can be naturally considered as a mapping defined by . The following theorem will show that is an L-fuzzy convexity on an effect algebra E.
Theorem 5. Let E be an effect algebra. Then is an L-fuzzy convexity on E.
Proof. (LMC1). It is clear that by Definition 8.
(LMC2). Let
be nonempty. Now we show
. Let
with
. Then for any
, we have
, which implies
and
for all
satisfying
. It follows that
This gives that . Hence .
(LMC3) Let
be nonempty and directed. Now, we show
. Let
with
. Then for any
,
implies
and
for all
satisfying
. Now, we prove
Take
with
then there exists
such that
,
and
. Since
is directed, there exists
such that
and
, we thus have
. By the fact that
, so
. Hence
Hence , it implies .
Therefore, is an L-fuzzy convexity. □
In order to investigate L-fuzzy convexity preserving mappings and L-fuzzy convex-to-convex mappings, we first give following definition and lemma.
Definition 11 ([
40]).
Let E and F be effect algebras and g: is called- (i)
a morphism if , and implies in F, and ;
- (ii)
a monomorphism if g is a morphism and iff .
Lemma 2. Let E and F be effect algebras and g: is a morphism. Then and for all
Proof. Since by Definition 11, it follows from Definition 1 (E3) that . Since for any , we have , thus . □
Theorem 6. Let be a morphism between two effect algebras E and F. Then
- (i)
g: is L-fuzzy convexity preserving.
- (ii)
If is a monomorphism, then g: is L-fuzzy convex-to-convex.
Proof. (i) In order to prove g: is L-fuzzy convexity preserving, we just need to prove for any , .
Let
with
. Then by Lemma 1 we obtain
and
for all
satisfying
. So for any
with
,
Similarly, we can obtain and , which implies . Hence .
(ii) In order to prove g: is L-fuzzy convex-to-convex, we just need to prove for any , .
Let
with
, then
and
for all
satisfying
. It follows that for
satisfying
,
Similarly, we can prove . This gives that . Hence . Therefore, is L-fuzzy convex-to-convex. □
Corollary 5. Let be a family of effect algebras and be the direct product of , where . If is the projection, then : is L-fuzzy convexity preserving for all .
Proof. It is easy to check that is a morphism between effects algebras, so by Theorem 6, we know is L-fuzzy convexity preserving for all . □
Theorem 7. Let be a family of effect algebras and be the direct product of , where . Then .
Proof. Let
be the projection. It can be easily proved that
. So by the proof of
(LMC2) in Theorem 8, we have
The last inequality holds because is L-fuzzy convexity preserving by Corollary 4. □
From paper [
27], we know
is an
L-convex space for all
. In particular,
is an
L-convex space. We should note that
, by Definition 9, we obtain that
is the set of all
L-subalgebras on an effect algebra
E.
The
L-convex hull operator with respect to the
L-convex space
is as follows: for
,
[
17], i.e.,
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 , Then .
Proof. We first show is an L-subalgebra of E. To achieve this, we give the following statements.
(i). For any and , .
Let
. For any
, it follows from
that
Let
. For any
, since
, we have that for each
.
Since for any
,
then for every pair
satisfying
, it holds that
and
, we thus obtain
.
(ii). .
Since
and
, we have
. By the above proof, we obtain
. This implies
(iii).
For any , from the construction of , it is clear that .
Let
. For any
, by the fact that
and (i), we have
(iv). .
For any
and
, since
, we have
By (i) we know that
, it follows that
. Thus
. By (iii) we obtain
(v). For
and
,
Let
such that
, then we have
and
. So there exist
such that
and
. Put
, then by (iii) we obtain
and
. It results that
Hence
(vi). .
By (ii) and (iv), we have,
Therefore, by (iii), (iv), (v), (vi) and Proposition 1, we can obtain that is an L-subalgebra containing A. Next, we prove is the least L-subalgebra containing A.
Assume
is an
L-subalgebra containing
A, so
for all
. It is clear that
. Since
Assume
holds. Now,
Thus for any , it holds that , which implies . This implies is the least L-subalgebra containing A. Therefore, □