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Closure Systems Over Effect Algebras Sistemas De Cierre Sobre Algebras

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Closure Systems Over Effect Algebras Sistemas De Cierre Sobre Algebras 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. * [email protected] (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 733 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). 734 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 735 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 736 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.
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