ISSN-E 1995-9516 Universidad Nacional de Ingeniería COPYRIGHT © (UNI). TODOS LOS DERECHOS RESERVADOS http://revistas.uni.edu.ni/index.php/Nexo https://doi.org/10.5377/nexo.v34i02.11558 Vol. 34, No. 02, pp. 733-743/Junio 2021
Closure systems over effect algebras
Sistemas de cierre sobre algebras de efecto
Mahdi Ronasi1, Esfandiar Eslami2*
1Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran. 2Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran.
(recibido/received: 03-diciembre-2020; aceptado/accepted: 15-abril-2021)
ABSTRACT
The present paper is an attempt to introduce the closure systems over effect algebras. At first, we will define closure systems over effect algebras, and for arbitrary set $ U $ and arbitrary subset S of all functions from U to an effect algebra L we will obtain the closure system containing S. Then, we will define the base of this closure system, and for arbitrary subset S of all functions from U to an effect algebra L we will obtain the base of this closure system.
Keywords: Closure Systems, Closure operator, Effect Algebra, Base.
RESUMEN
El presente artículo es un intento de introducir los sistemas de cierre sobre las álgebras de efectos. Primero definiremos sistemas de cierre sobre álgebras de efectos y para el conjunto arbitrario $ U $ y el subconjunto arbitrario S de todas las funciones de U a un álgebra de efectos L obtendremos el sistema de cierre que contiene S. Luego definiremos la base de este sistema de cierre y para un subconjunto arbitrario S de todas las funciones desde U hasta un álgebra de efectos L obtendremos la base de este sistema de cierre.
Palabras clave: Sistemas de cierre, Operador de cierre, Álgebra de efectos, Base.
1. INTRODUCTION
Closure systems have an important role in almost all parts of mathematics and computer science, specially databases, data analysis and management of data. The first people who introduced the concept of closure system were (Abramsky & Jung, 1994), and (Belohlavek, 2001). L closure systems (with L being a partially ordered set) are introduced in (Abramsky & Jung, 1994) and (Belohlavek, 2001) and (Nola et al., 2002) and (Lu & Wang, 2011) This concept generalizes the
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ordinary subsets of U or equivalently, characteristic functions A : U → {0, 1} to A : U →L that are referred to as L -sets. The partially ordered set L can have different algebraic structures such as Boolean algebra, MV-algebra, BL-algebra or effect algebra Effect algebra was first introduced by (Foulis & Bennett, 1994) An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we try to introduce the concept closure system over effect algebra. This paper is organized as following: In section 2, we will present the content that we need in the paper. In section 3, we will introduce closure system over effect algebra. In section 4, we will introduce bases of a closure system.
2. MATERIALS AND METHODS
For research in this article, library studies and a collection of articles and books have been used.
Preliminaries Effect algebras are abstract generalizations of the unit interval [0,1]⊆ ℝ. [0,1]⊆ ℝ carries a partial addition: the sum of two elements may or may not lie in the [0,1]⊆ ℝ again. Furthermore, it has a minimal and a maximal element, and complements with respect to the maximal element. We capture the algebraic structure of [0,1] in the notion of an effect algebra 2.1. Definition (Foulis & Bennett, 1994) An effect algebra L is a structure (L, +, ´, 0, 1) consisting of a set L with two special elements 0, 1, unary operation ´ and a partially defined binary operation + on 퐿 × 퐿 satisfying the following conditions for every 푥, 푦 , 푧 ∈ 퐿 1. Commutativity: if 푥 + 푦 is defined, then so is 푦 + 푥, and 푥 + 푦 = 푦 + 푥 2. Associativity: if 푥 + 푦 and (푥 + 푦) + 푧 are defined, then so are 푦 + 푧 and 푥 + (푦 + 푧), and (푥 + 푦) + 푧 = 푥 + (푦 + 푧). 3. Zero: 0+a is always defined and equals a 4. Orthocomplement: for each 푥 ∈ 퐿, 푥´ is the unique element for which 푥 + 푥´ = 1 5. Zero-one law: if 푥 + 1 is defined, then 푥 = 0. When 푥 + 푦 exists, we say that 푥 is orthogonal to y and will denote this by 푥 ⊥ 푦.
On an effect algebra L one can define a partial order ≤ as (O1) 푥 ≤ 푦 ⟺ ∃ 푡 ∈ 퐿 (푥 + 푡 = 푦)
Below we list a useful set of properties of effect algebras (Foulis & Bennett, 1994) and (Dvurecenskij & Pulmannova, 2000).
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2.2. Proposition Let 퐿 be an effect algebra and 푎, 푏 and 푐 be elements of 퐿:
´´ (P1) a = a. (1) ´ ´ (P2) 1 = 0, 0 = 1. (2) (P3) a ⊥ 0 , a + 0 = 1. (3) (P4) a ⊥ 1 ⟺ a = 0. (4) (P5) a + b = 0 ⟺ a = b = 0. (5) ´ (P6) a ⊥ b ⟺ a ≤ b . (6) ´ ´ (P7) a ≤ b ⟺ b ≤ a . (7) (P8) a + c = b + c ⇒ a = b. (8) (P9) a + c ≤ b + c ⇒ a ≤ b. (9) ´ (P10) a ≤ b ⇒ a ⊥ (a + b´) . (10) ´ (P11) a ≤ b ⇒ a ⊥ (a + b´) = b. (11) (P12) a + b = c ⟺ a´ = b + c´. (12)
2.3. Lemma
In (푃8) 푎푛푑 ( 푃9) of the previous proposition, if 푐 ⊥ 푎 , 푐 ⊥ 푏, it can be concluded that
(푃8´) 푎 = 푏 ⇒ 푎 + 푐 = 푏 + 푐 (13) (푃9´) 푎 ≤ 푏 ⇒ 푎 + 푐 ≤ 푏 + 푐 (14)
Proof:
(푃8´) It is clear.
(푃9´) 푆푖푛푐푒 푎 ≤ 푏 푏푦 ( 푂1) 푡ℎ푒푟푒 푒푥푖푠푡푠 푡 ∈ 퐿, 푏 = 푎 + 푡, 푠표 푐 + 푏 = 푐 + 푎 + 푡 . 푇ℎ푒푟푒푓표푟푒 , 푎 + 푐 ≤ 푏 + 푐. A closure system 푆 on a set 퐿 is a set of subsets of 퐿 containing 퐿 and any intersection of subsets of 푆. In the sequel we can see the definition of closure system over effect algebras. If 푈 is a set and 퐿 is an effect algebra, then we define 퐿푈 as the set of all functions from 푈 to 퐿 An 퐿 - closure operator in 푈 is a function 퐹: 퐿푈 ⟶ 퐿푈 such that for all 퐴 , 퐵 ∈ 퐿푈 the following properties hold 퐴 ⊆ 퐹(퐴) If 퐴 ⊆ 퐵 then, 퐹(퐴) ⊆ 퐹(퐵) 퐹(퐹(퐴)) = 퐴
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3. RESULTS AND DISCUSSION
If 퐿 be an effect algebra and 푈 be a nonempty set. For 푆 ⊆ 퐿푈, the closure system which containing 푆 is 푆∞ and 푅푑푢+(. . . 푅푑푢+(푅푑푢⋀(푅푑푢+(푆)))) or 푅푑푢⋀(. . . 푅푑푢⋀(푅푑푢+(푅푑푢⋀(푆)))) is a base of 푆∞ .
Closure Systems Over Effect Algebras 3.1. Definition If 퐿 is an effect algebra, 푈 is a nonempty set and 푆 ⊆ 퐿푈we call 푆 a closure system over 퐿 if the following conditions are satisfied:
푆 is closed under ⋀ − intersections, i.e. if 퐴푖 ∈ 푆 ,then ⋀퐴푖 ∈ 푆. 푆 is closed under summations, i.e. for all 퐴 ∈ 푆 and 푎 ∈ 퐿, if 푎 ⊥ 퐴 , then 푎 + 퐴 ∈ 푆 . Here, 푎 + 퐴 , ⋀퐴푖 are defined by
(푎 + 퐴)(푢) = 푎 + 퐴(푢), (⋀퐴푖) = ⋀퐴푖(푢). (15)
In which 푎 ⊥ 퐴 means , 푎 ⊥ 퐴(푢) for all 푢 ∈ 푈. Obviously, the set 퐿푈 is a closure system (the largest one) and we can easily see that an intersection of an arbitrary system of closure systems is a closure system. Hence, it follows from classical results (Davey, 2002) that for every set 푆 ⊆ 퐿푈 there exists the least closure system ̅푆 containing 푆, namely the intersection of all closure systems that contain 푆. 3.2. Definition A base of a closure system 푇 ⊆ 퐿푈 is a set 푆 ⊆ 퐿푈 such that 푆̅ = 푇. 푃̅ ≠ 푇, for every 푃 ⊆ 푆. 3.3. Definition For 푆 ⊆ 퐿푈, we put: 푆̂ = {⋀퐴 ∶ ∅ ≠ 퐴 ⊆ 푆} (16) 푆+ = {푎 + 퐴 ∶ 푎 ∈ 퐿 , 퐴 ∈ 푆, 푎 ⊥ 퐴} (17)
푎 푎 in the unit interval [0,1] ⊆ ℝ We mean the phrase for the rational number, i.e. ∈ ℚ $ and 푏 푏 푎 푐 + is partial addition in [0,1] ⊆ ℝ , i.e. + may or may not lie in the [0,1]⊆ ℝ again. In 푏 푑 the next example (푎, 푏 ) or [푎, 푏] means a set of all functions whose values are in the (푎, 푏) or [푎, 푏], in which (푎, 푏) and [푎, 푏] are open and closed intervals in ℝ
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3.4. Example consider the unit interval [0,1] ⊆ ℝ , 푈 = {1} and
1 1 2 (a) 푆 = { , , } 1 5 8 7 1 1 2 1 Hence 푆̂ = 푆 , 푆 = [ , 1] ∪ [ , 1] ∪ [ , 1] = [ , 1] 1 1 1+ 5 8 7 8 1 1 3 8 (b) 푆 = { , ( , ) , } 1 10 7 4 9
1 Then 푆̂ = 푆 , 푆 = 푆̂ =[ , 1] 2 2 2+ 2+ 10
It is well known that the closure systems and closure operators are cryptomorphic 1 mathematical structures\\. the closure operator associated with a closure system defines the closure of a subset 퐸 of 퐿 as the least closed set containing 퐸 and the closure system associated with a closure operator is the family of its fixed points. in the next theorem we show that functions ̂ 푈 푈 (. ) , (. )+: 풫(퐿 ) ⟶ 풫(퐿 ) such that for all 푈 ̂ 푆 ⊆ 퐿 , (. ) (푆) = 푆̂, (. )+(푆) = 푆+ are closure operators. 3.5. Theorem ̂ Let 퐿 be an effect algebra and 푈 be a nonempty set, then (. ) , (. )+ are closure operators on 퐿푈. Proof. It is clear that for all 푆 ⊆ 퐿푈, 푆 ⊆ 푆̂ . ̂ Let 푆1 ⊆ 푆2. Since for all 푇 ⊆ 푆1, 푇 ⊆ 푆2 , we have 푆̂1 ⊆ 푆̂2. Now consider 푆̂ = {⋀푇: 푇 ⊆ 푆̂} (I)Informally a structure Γ on a set 퐸 can be seen as a set of axioms bearing on mathematical objects(operations, maps, families of subsets,..) defined on 퐸 Let Γ , 훤 ´ be two structures defined on 퐸. They are cryptomorphic if there exist maps between the objects of the two structures which transform any assertion true in one of these structures into an assertion true in the other one. For instance, the structure of Boolean algebra is cryptomorphic with the structure of Boolean ring (Nicoletti et al., 1988), for a more precise formulation.
푈 Since for all 푆 ⊆ 퐿 , 푆 = 0 + 푆. Hence 푆 ⊆ 푆+ . Now consider 푎 + 퐴 ∈ 푆++ hence 푎 ∈ 푆+, 푎 ⊥ 푡 , for all t ∈A . Since 퐴 ∈ 푆+ , therefore 퐴 = 푎´ + 퐴 ´, which 퐴 ´ ∈ 푆, 푎´ ⊥ 푡 for all 푡 ∈ 퐴´ . So 푎 + 퐴 = 푎 + 푎´ + 퐴´ ∈ 푆+ .
Now let 푆1 ⊆ 푆2 , 퐴 ∈ 푆1 and 푎 + 퐴 ⊆ 푆1+ ⊆ 푆2+, so (. )+ is a closure operator.
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We may define the direct product of a family of effect algebras as follows Assume that 푈 is a nonempty set and 퐿푢, 푢 ∈ 푈 are effect algebras. The direct product of family { 퐿푢: 푢 ∈ 푈 } of effect algebras , denoted by ∏푢∈푈 퐿푢 is an effect algebra in which partially binary operation ´ + and unary operation (−) defined pointwise. In the other words ∏푢∈푈 퐿푢 is the set of all functions 푓: 푈 ⟶ ⋃푢∈푈 퐿푢 such that 푓(푢) ∈ 퐿푢 for all 푢 ∈ 푈 with the partially defined binary operation +and the unary operation (−)´defined by: if for all 푢 ∈ 푈, 푓(푢) ⊥ 푔(푢), (푓 + 푔)(푢) = 푓(푢) + 푔(푢), 푓´(푢) = 푓(푢)´ . The least and the greatest element of ∏푢∈푈 퐿푢 are the functions 0,1: 푈 ⟶ ⋃푢∈푈 퐿푢 such that
0(푢) = 0퐿푢, 1(푢) = 1퐿푢.
Order in direct product is pointwise, i.e. ∏푢∈푈 푎푢 ≤ ∏푢∈푈 푏푢 iff 푎푢 ≤ 푏푢. Since operations in direct product of effect algebras are pointwise, therefore direct product of effect algebras is an effect algebra. 3.6. Proposition The direct product of effect algebras is an effect algebra Like MV-algebras (Nola et al., 2002) for every effect algebra 퐿 and nonempty set 푈, 퐿푢 is direct product of the family
{ 퐿푢: 푢 ∈ 푈 } where 퐿푢 = 퐿 , for all 푢 ∈ 푈. Based on what has been said, the following examples are subsets of direct product of [0,1]
The following example shows that in general, even if 퐿 is a chain 푆̂+ ⊈ 푆̂+ , 푆̂+ ⊈ 푆̂+ 1 1 In the next example by ( , ) we mean a function like 푓 from {0,1} to [0,1] in which 1000 20 1 1 푓(0) = , 푓(1) = 1000 20 3.7. Example Consider the unit interval [0, 1] ⊆ ℝ, 푈 = {0, 1}
1 1 1 1 (1) 푆 = {( , ) , ( , )} 1 1000 20 40 300 1 1 1 1 1 1 1 1 1 1 Obviously ( + , + ) ⋀ ( , ) = ( + , ) ∈ 푆̂ , but ( + 1000 1000 1000 20 40 300 1000 1000 300 1+ 1000 1 1 , ) ∉ 푆̂ . So in general 1000 300 1+
̂ ̂ 푆1+ ⊈ 푆1+ 1 99 9 1 (2) 푆 = {( , ) , ( , )} 2 1000 100 10 100 99 1 1 99 1 Therefore + ( , ) = ( + , 1) ∈ 푆̂ . 100 1000 100 100 1000 2+
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99 1 99 1 1 99 9 We claim ( + , 1) ∉ 푆̂ , if ( + , 1) = (푡 + , 푡 + ) ⋀ ( 푡´ + , 푡´ + 100 1000 2+ 100 1000 1000 100 10 1 1 1 ) = (푡 + , 푡´ + ) 100 1000 100 1 Where 0 ≤ 푡´ ≤ , which is a contradiction. 10 3.8. Lemma
푈 ̂ Let 퐿 be an effect algebra and 푈 be a nonempty set, 푆 ⊆ 퐿 , 푆̅0 = 푆 and 푆̅̅푖̅+̅1̅ = 푆̅푖 ∪ ̅ ̅ 푆푖+ then ⋃푆푖 is a closure system.
Proof. Let ∈ ⋃푆푖 . So for some 푖, 퐴 ∈ 푆푖 . Now we consider 푎 ∈ 퐿 in which, 푎 ⊥ 퐴, then ̅ ̅̅̅̅̅ ̅ 푎 + 퐴 ∈ 푆푖+ ⊆ 푆푖+1 ⊆ ⋃푆푖 .
Now let {퐴푖: 푖 ∈ 퐼} ⊆ ⋃푆̅푖 , since 푆̅푖 ⊆ 푆̅̅푖̅+̅1̅ , there exists ̂ 푗 such that {퐴푖: 푖 ∈ 퐼} ⊆ 푆̅푗. Therefore ⋀퐴푖 ∈ 푆̅푗 ⊆ 푆̅̅푗̅+̅1̅ ⊆ ⋃푆̅푖 . 3.9. Corollary
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set, 푆 ⊆ 퐿 , then 푆̅ = ⋃푆̅푖 where, , ̅ ̅̅̅̅̅ ̅̂ ̅ 푆0 = 푆 and 푆푖+1 = 푆푖 ∪ 푆푖+. ̅̂ ̅ ̅ ̅ Proof. Since for all , 푆푖 ⊆ 푆, 푆푖+ ⊆ 푆 . Therefore
⋃푆̅푖 ⊆ 푆.̅ On the other hand , according to Lemma, because ⋃푆̅푖 is a closure system and ̅푆 is the smallest closure system, therefore 푆̅ ⊆ ⋃푆̅푖. 3.10. Definition Let 퐿 be an effect algebra and 푈 be a nonempty set, for each 푆 ⊆ 퐿푈 , we define
푆⊕ = {푎푢 + 퐴(푢): 푎푢 ∈ 퐿 , 퐴 ∈ 푆 , 푎푢 ⊥ 퐴(푢)} (18)
The next example shows that in general 푆+ ≠ 푆⊕. 3.11. Example Consider = [0,1], 푈 = {0,1} , 1 99 9 1 1 99 1 9 1 푆 = {( , ),( , )}, so 푆 = {( + 푡, + 푡) : 0 ≤ 푡 ≤ } ∪ {( + 푡, + 푡) : 0 ≤ 1000 100 10 100 + 100 100 100 10 100 1 푡 ≤ }. 10 1 99 999 1 9 1 But 푆 = {( + 푡, + 푡´) : 0 ≤ 푡 ≤ , 0 ≤ 푡´ ≤ } ∪ {( + 푡, + 푡´) : 0 ≤ 푡 ≤ ⊕ 1000 100 1000 100 10 100 1 99 , 0 ≤ 푡´ ≤ } 10 100 3.12. Definition Let 퐿 be an effect algebra and 푈 be a nonempty set and 푆 ⊆ 퐿푈 , we define
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… ̂̂ (19) 푆∞ = 푆++ …
3.13. Lemma
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set and 푆 ⊆ 퐿 , then 푆̅ = 푆∞. Proof. Since for all
퐴 ⊆ 푆∞. 퐴̂ , 퐴+ ⊆ 푆∞ , so 푆∞ is a closure system . Therefore 푆̅ ⊆ 푆∞. Since 푆̂ ⊆ 푆̅̅1̅̅ , 푆̂+ ⊆ ̅푆̅̅2̅ , … we have 푆∞ ⊆ 푆.̅
3.14. Corollary Let 퐿 be an effect algebra and 푈 be a nonempty set and 푆 ⊆ 퐿푈, then ̂ … 푆 = ̂̂푆 (20) ∞ +++ …
̂ … ̂ … Proof. Since ̂̂푆 is a closure system , 푆̅ ⊆ ̂̂푆 , and since 푆 ⊆ 푆̅̅̅, 푆̂ ⊆ +++ … +++ … + 1 + ̂ … 푆̅̅̅, …, we have ̂̂푆 ⊆ 푆̅ 2 +++ …
(̂. )- Base , (. )+- Base and (̅̅.̅̅)- Base In this section, we are giving some algorithms to construct different bases. First some basic concepts are reviewed 3.15. Definition (Belohlavek & Konecny, 2016) Let 퐿 be an effect algebra and 푈 be a nonempty set. For 푆 ⊆ 퐿푈, (̂. )- base of 푆̂ is a set 푈 푆0 ⊆ 퐿 such that
푆̂0 = 푆̂ 푇̂ ≠ 푆̂ for every 푇 ⊂ 푆0
Let 퐿 be an effect algebra and 푈 be a nonempty set. For 푆 ⊆ 퐿푈 we consider the set of elements in 푆 minimal with respect to (̂. ) , i.e. ̂ 푅푑푢⋀(푆) = {퐴 ∈ 푆 ∶ 퐴 ∉ 푆 − {퐴}} The following theorem is a folklore in lattice theory. Note also that the theorem follows from the results on bases in domain theory on irreducibility (Abramsky & Jung, 1994) and (Gierz et al., 2004) and (Mundici, 2007).
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3.16. Theorem
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set. For every finite 푆 ⊆ 퐿 , 푅푑푢⋀(푆) is a unique (̂. )- base of 푆̂ .
3.17. Definition Let 퐿 be an effect algebra and 푈 be a nonempty set. Let 푅 denote the binary relation on 퐿푈 defined by
퐵1 푅 퐵2 if and only if, for some 푎 ∈ 퐿 , 푎 ⊥ 퐵1 and 퐵2 = 푎 + 퐵1.
3.18. Lemma Let 푅 be defined as above. Then 1. 푅 is reflexive, antisymmetric and transitive
2. 퐵1 푅 퐵2 implies, 퐵2+ ⊆ 퐵1+ .
Proof. 1. Since 0 + 퐴 = 퐴, 푅 is reflexive. Now let for some 푎 , 푎´ ∈ 퐿 and 퐵1 , 퐵2 ∈ 푆 , 푎 + 퐵1 = 퐵2 , 푎´ + 퐵2 = 퐵1 . So for all 푢 ∈ 푈 , 푎 + 퐵1(푢) = 퐵2 (푢), 푎´ + 퐵2 (푢) = 퐵1 (푢) . Hence 푎 + 푎´ = 0 , so by (푃5), 푎 = 푎´ = 0 , therefore 푅 is antisymmetric. Let for some 푎 , 푎´ ∈ 퐿 and 퐵1 , 퐵2 , 퐵3 ∈ 푆 . 퐵2 = 푎 + 퐵1, 퐵3 = 푎´ + 퐵2 , so 퐵3 = 푎´ + 푎 + 퐵2. Thus 푅 is transitive. 2. Since 퐵1 푅 퐵2 for some 푎 ∈ 퐿 , 퐵2 = 푎 + 퐵1. Now consider 푎´ + 퐵2 ∈ 퐵2+ , clearly 푎´ + 퐵2 = 푎´ + 푎 + 퐵1 ∈ 퐵1+ .
3.19. Definition
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set. For every finite 푆 ⊆ 퐿 , 푅푑푢+(푆) denote the set of all minimal elements in 푆 with respect to 푅 , i.e.
푅푑푢+(푆) = {퐵 ∈ 푆 ∶ 퐵1 푅 퐵 푖푚푝푙푖푒푠 퐵1 = 퐵 푓표푟 푠표푚푒 퐵1 ∈ 푆} (21)
3.20. Definition
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set. For 푆 ⊆ 퐿 , (. )+- base of 푆 푆+ is a set
푈 푆0 ⊆ 퐿 such that
푆0+ = 푆+
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푇+ ≠ 푆+ for every 푇 ⊂ 푆0
3.21. Theorem
푈 Let 퐿 be an effect algebra and 푈 be a nonempty set. For any finite 푆 ⊆ 퐿 , 푅푑푢+(푆) is unique
(. )+- base of 푆+ Proof.
We prove at first (푅푑푢+(푆))+ = 푆+ . By definition, we have
(푅푑푢+(푆))+ = {푎 + 퐵 ∶ 푎 ⊥ 퐵, 퐵 ∈ 푅푑푢+(푆)}. Since
푅푑푢+(푆) ⊆ 푆 , (푅푑푢+(푆))+ ⊆ 푆+ . Now let 퐵 ∈ 푆+ .
So, 퐵 = 푎1 + 퐵1 for some 퐵1 ∈ 푆 , 푎1 ∈ 퐿.
Since 푆 is finite, there exists 퐵2 ∈ 푅푑푢+(푆) such that
퐵2 푅 퐵1 ,i.e. 퐵1 = 푎2 + 퐵2, for some 푎2 ∈ 퐿. Hence
퐵 = 푎1 + 퐵1 = 푎1 + (푎2 + 퐵2) = (푎1 + 푎2) + 퐵2 ∈ (푅푑푢+(푆))+
If 푅푑푢+(푆) is redundant, then there exists 퐵 ∈ 푅푑푢+(푆) such that , 퐵 ∈ ( 푅푑푢+(푆) − {퐵})+. Therefore 퐵 = 푎 + 퐵1 such that 퐵1 ≠ 퐵, which is a contradiction.
Now let 푇 is another (. )+- base . Then since 푅푑푢+(푆) ⊆ 푆+ = 푇+. For each 퐵 ∈ 푅푑푢+(푆) , there exists 푎1 ∈ 퐿 , 퐵1 ∈ 푇 such that 퐵 = 푎1 + 퐵1,i.e. 퐵1 푅 퐵 . As 퐵1 ∈ 푇 ⊆ 푇+ = 푆+, there exists 푎2 ∈ 퐿 and 퐵2 ∈ 푅푑푢+(푆) such that 퐵1 = 푎2 + 퐵2 ,i.e. 퐵2 푅 퐵1 .Due to transitivity of 푅 . Since 퐵, 퐵2 ∈ 푅푑푢+(푆) and since 퐵 is minimal in 푆, we obtain 퐵 = 퐵2. Observe that by previous Lemma we have 퐵2 ⊆ 퐵1 ⊆ 퐵 therefore 퐵 = 퐵1 ∈ 푇 . Therefore 푅푑푢+(푆) ⊆ 푇 . since 푅푑푢+(푆) is a (. )+- base, we must have because 푅푑푢+(푆) = 푇 , otherwise 푇 is redundant.
3. CONCLUSION
푈 Lemma Let 퐿 be an effect algebra and 푈 be a nonempty set , 푆1, 푆2, 푆3, … ⊆ 퐿 , if 푆1+ = 푆2+, ̂ ̂ 푆2 = 푆3 , 푆3+ = 푆4+, …
Then 푆̅1 = 푆̅2 = 푆̅3 = 푆̅4 = ⋯ Proof. By Lemma 3.13 and Corollary 3.14 … … … … ̂̂ ̂̂ ̂̂ ̂̂ (22) 푆̅1 = 푆1 = ̂ 푆1 … = ̂ 푆2 … = 푆̅̅ퟐ̅ = 푆̂2 + + … = ̂푆3 + + … = 푆̅̅ퟑ̅ = ⋯ ∞ +++ +++
푈 Lemma Let 퐿 be an effect algebra and 푈 be a nonempty set , 푆1, 푆2, 푆3, … ⊆ 퐿 , if 푆̂1 = ̂ ̂ ̂ 푆2 , 푆2+ = 푆3+, 푆3 = 푆4 , …
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Then, 푆̅1 = 푆̅2 = 푆̅3 = 푆̅4 = ⋯ Proof. Similar to the previous Lemma. The following result is obtained from Lemma 4.1
If 푆1 is the input system of 퐿-sets, we obtain (a smaller ) 푆2 with 푆1+ = 푆2+ and then obtain ̂ ̂ from 푆2 some smaller 푆3 with 푆2 = 푆3 and we obtain from 푆3 some smaller 푆4 with 푆3+ =
푆4+,... then 푆1, 푆2, 푆3, 푆4, … generate the same 퐿 -closure systems. In particular, in view of the preceding results, a good choice is to take 푅푑푢+(푆), 푅푑푢⋀(푆) , In this procedure, These considerations bring us a way to find the base of 푆.̅ Corollary Let 퐿 be an effect algebra and 푈 be a nonempty set. If 푆 ⊆ 퐿푈, then 푅푑푢+(. . . 푅푑푢+(푅푑푢⋀(푅푑푢+(푆)))) or 푅푑푢⋀(. . . 푅푑푢⋀(푅푑푢+(푅푑푢⋀(푆)))) is a base of 푆.̅ future work In future work, we are looking for conditions in which 푆 ̅ and bass for 푆 ̅ can be operated with finite operations.
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