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LA-SEMIGROUPS CHARACTERIZED BY THE PROPERTIES OF INTERVAL VALUED (α, β)-FUZZY IDEALS
LA-SEMIGROUPS CHARACTERIZED BY THE PROPERTIES OF INTERVAL VALUED (α, β)-FUZZY IDEALS
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 405-426
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : October 31, 2012
  • Accepted : February 08, 2014
  • Published : September 28, 2014
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About the Authors
SALEEM ABDULLAH
SAMREEN ASLAM
NOOR UL AMIN

Abstract
The concept of interval-valued ( α, β )-fuzzy ideals, intervalvalued ( α, β )-fuzzy generalized bi-ideals are introduced in LA-semigroups, using the ideas of belonging and quasi-coincidence of an interval-valued fuzzy point with an interval-valued fuzzy set and some related properties are investigated. We define the lower and upper parts of interval-valued fuzzy subsets of an LA-semigroup. Regular LA-semigroups are characterized by the properties of the lower part of interval-valued (∈, ∈, ˅ q )-fuzzy left ideals, interval-valued (∈, ∈, ˅ q )-fuzzy quasi-ideals and interval-valued (∈, ∈, ˅ q )-fuzzy generalized bi-ideals. Main Facts . AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
The concept of fuzzy sets was first introduced by Zadeh [16] and then the fuzzy sets have been used in the reconsideration of classical mathematics. Fuzzy set theory has been shown to be a useful tool to describe situations in which data is imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoratical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. The notion of fuzzy subgroups was defined by Rosen ed [12] . A systematic exposition of fuzzy semigroup was given by Morde-son et. al. [7] , and they have find theoratical results on fuzzy semigroups and their use in finite state machine, fuzzy languages and fuzzy coding. Using the notions "belong to" relation (∈) introduced by Pu and Liu [11] , in [8] Morali proposed the concept of a fuzzy point belonging to a fuzzy subset under nat-ural equivalence on fuzzy subsets. Bhakat and Das introduced the concept of ( α, β )-fuzzy subgroups by using the belong to relation (∈) and quasi-coincident with relation (q) between a fuzzy point and a fuzzy subgroup, and defined an (∈, ∈ ∨ q )-fuzzy subgroup of a group [1] . Kazanci and Yamak [4] studied gener-alized types fuzzy bi-ideals of semigroups and defined
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-fuzzy bi-ideals of semigroups. In [14] Shabir et. al. characterized regular semigroups by the properties of ( α, β )-fuzzy ideals, bi-ideals and quasi-ideals. In [15] , Shabir and Yasir characterized regular semigroups by the properties of
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-fuzzy ideals, generalized bi-ideals and quasi-ideals of a semigroup.
Interval-valued fuzzy subsets were proposed about thirty years ago as a natu-ral extension of fuzzy sets by Zadeh [17] . Interval-valued fuzzy subsets have many applications in several areas such as the method of approximate inference. In [10] Al Narayanan and T. Manikantan introduced the notions of interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in semigroups.
In this paper we introduced the concepts of interval-valued (∈, ∈ ∨ q )-fuzzy ideals, interval-valued (∈,∈ ∨ q )-fuzzy bi-ideals and interval-value (∈,∈ ∨ ∧ q )-fuzzy quasi-ideals of an LA-semigroup, where α, β are any one of {∈, q , ∈ ∨ q , ∈ ∧ q } with α ≠ ∈ ∧ q , by using belong to relation (∈) and quasi-coincidence with relation ( q ) between interval-valued fuzzy set and an interval-valued fuzzy point, and investigated related properties.
2. Preliminaries
We first recall some basic concepts. A groupoid ( S , ∗) is called a left almost semigroup, abbreviated as an LA-semigroup, if it satisfies left invertive law:
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A non-empty subset A of S is called a sub LA-semigroup of S if AA A and is called left (resp. right) ideal of S if SA A ( AS A ). By a two-sided ideal or simply an ideal we mean a non-empty subset of S which is both a left and a right ideal of S . A non-empty subset B of S is called a generalized bi-ideal of S if ( BS ) B B . A sub LA-semigroup B of S is called a bi-ideal of S if ( BS ) B B . A non-empty subset Q of S is called a quasi-ideal of S if QS SQ Q . Obviously every one-sided ideal of an LA-semigroup S is a quasi-ideal, every quasi-ideal is a bi-ideal and every bi-ideal is a generalized bi-ideal. An LA-semigroup S is called regular if for each element a of S , there exists an element x in S such that a = ( ax ) a . It is well known that for a regular LA-semigroup the concepts of generalized bi-ideal, bi-ideal and quasi-ideal coincide.
We now review some concepts of fuzzy subsets. A fuzzy subset λ of an LA-semigroup S is a mapping λ : S → [0, 1]. Throughout this paper S denotes an LA-semigroup.
Definition 1. A fuzzy subset λ of S is called a fuzzy sub LA-semigroup of S if for all x, y S
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Definition 2. A fuzzy subset λ of S is called a fuzzy left (resp. right) ideal of S if for all x, y S
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A fuzzy subset λ of S is called a fuzzy two-sided ideal or simply fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S .
Definition 3. A fuzzy subsemigroup λ of S is called a fuzzy bi-ideal of S if for all x, y S
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We will describe some results of an interval number. By an interval number on [0, 1], say ã is a closed subinterval of [0, 1], that is, ã = [a , a + ], where 0 ≤ a ≤ a + ≤ 1. Let D [0, 1] denote the family of all closed subintervals of
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Now we define ≤,=, ∧, ∨ in case of two elements in D [0, 1].
Consider two elements ã = [a , a + ], and
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in D [0, 1]. Then,
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if and only if a ≤ b and a + ≤ b + .
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if and only if a = b and a + = b + .
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= [min{a , b }, min{a + , b + }].
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= [max{a , b }, max{a + , b + }].
Let X be a set. A mapping
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: X D [0, 1] is called an interval-valued fuzzy subset ( briey, an i-v fuzzy subset) of X , where
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for all x X , where λ and λ + are fuzzy subsets in X such that λ ( x ) ≤ λ + ( x ) for all x X .
Let
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be two interval-valued fuzzy subsets of X . Define the relation ⊆ between λ and μ as follows:
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if and only if
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for all x X , that is, λ ( x ) ≤ μ ( x ) and λ + ( x ) ≤ μ + ( x ). An interval-valued fuzzy subset
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in a universe X of the form
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for all y X , is said to be an interval-valued fuzzy point with support x and interval value
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and is denoted by
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J. Zhan et al [19] gave meaning to the symbol
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where α ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }. An interval-valued fuzzy point
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is said to belongs to (resp. quasi-coincident with ) an interval-valued fuzzy set
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written
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if
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and in this case
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means that
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To say that
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means that
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does not hold.
Let
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be two interval-valued fuzzy subsets of an LA-semigroup S . The product
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is defined by
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3. Main results
Definition 4. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued fuzzy sub LA-semigroup of S if for all x, y S
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Definition 5. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued fuzzy left (resp. right ) ideal of S if for all x, y S
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Definition 6. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued fuzzy generalized bi-ideal of S if for all x, y, z S
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Definition 7. An interval-valued fuzzy sub LA-semigroup
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of S is called an interval-valued fuzzy bi-ideal of S if for all x, y, z S
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Definition 8. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued fuzzy quasi-ideal of S if
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where
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: S → [1, 1].
Theorem 1 ( [18] ). Let S be an LA - semigroup with left identity e such that ( xe ) S = xS for all x S. Then, the following are equivalent
  • (1)Sis regular.
  • (2)R∩L=RLfor every right idealRand left idealLofS.
  • (3)A= (AS)Afor every quasi-idealAofS.
4. INTERVAL-VALUED (α, β)-FUZZY IDEALS OF LA-SEMIGROUPS
Let S be an LA-semigroup and α, β denote any one of ∈, q , ∈ ∨ q , or ∈ ∧ q unless otherwise specified.
Definition 9. An interval-valued fuzzy subsete
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an LA-semigroup S is called an interval-valued ( α, β )-fuzzy sub LA-semigroup of S , where α ≠ ∈ ∧ q , if
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implies that
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Let
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be an interval-valued fuzzy subset of S such that,
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for all x S . Let x S and
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D [0, 1] such that,
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that is,
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It follows that
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This means that
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Therefore, the case α =∈ ∧ q in above definition is omitted.
Definition 10. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued ( α, β )-fuzzy left (resp. right ) ideal of S , where α ≠ ∈ ∧ q , if it satisfies,
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and x S implies that
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for all x, y S .
An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued ( α, β )-fuzzy ideal of S if it is both an interval-valued ( α, β )-fuzzy left ideal and an interval-valued ( α, β )-fuzzy right ideal of S .
Definition 11. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued ( α, β )-fuzzy generalized bi-ideal of S , where α ≠ ∈ ∧ q , if it satisfies
for all x, y, z S and for alle
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where
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implies that
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Definition 12. An interval-valued fuzzy subset
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of an LA-semigroup S is called an interval-valued ( α, β )-fuzzy bi-ideal of S , where α ≠ ∈ ∧ q , if it satisfies, the following two conditions.
(i) For all x, y S and for all
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where
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and
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implies that
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(ii) For all x, y, z S and for all
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where
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and
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implies that
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Lemma 1. An interval - valued fuzzy subset
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of an LA - semigroup S is an interval - valued fuzzy sub LA - semigroup of S if and only if it satisfies,
For all x, y S and for all
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such that
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implies that
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Proof . Suppose that
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is an interval-valued fuzzy sub LA-semigroup of an LA-semigroup S . Let x, y S and
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where
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such that,
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Then
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Since
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is an interval-valued fuzzy sub LA-semigroup of S . So
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Hence,
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Conversely, assume that
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satisfies the given condition. We show that
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On contrary, assume that there exist x, y S such that
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Let
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such that
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Then
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This contradicts our hypothesis. Thus,
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Lemma 2. An interval - valued fuzzy subset
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of an LA - semigroup S is an interval-valued fuzzy left ( resp. right ) ideal of S if and only if it satisfies
for all x, y S and for all
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where
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such that
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implies that
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Proof . Suppose that
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is an interval-valued fuzzy left ideal of an LA-semigroup S . If
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Since
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is an interval-valued fuzzy left ideal of S , so
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Hence,
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Conversely, suppose that
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satisfies the given condition. We show that
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On contrary, assume that there exist x, y S such that
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Let
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be such that
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Then
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Which contradicts our hypothesis. Hence,
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Remark 1. The above lemma shows that every fuzzy left (resp. right) ideal of S is an (∈, ∈)-fuzzy left (resp. right) ideal of S .
Lemma 3. An interval - valued fuzzy subset
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of an LA - semigroup S is an interval - valued fuzzy generalized bi-ideal of S if and only if it satisfies,
For all x, y, z S and for all
PPT Slide
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where
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implies that
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Proof . Suppose that
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is an interval-valued fuzzy generalized bi-ideal of S . Let x, y, z S and
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where
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such that
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Then,
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Since
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is an interval-valued fuzzy gener-alized bi-ideal of S . So
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Hence,
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Conversely, suppose that
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satisfies the given condition. We show that
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Suppose contrary that
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Let
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be such that
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Then,
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Which contradicts our supposition. Hence,
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Lemma 4. An interval - valued fuzzy subset
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of an LA - semigroup S is an interval - valued fuzzy bi - ideal of S if and only if it satisfy,
(1) for all x, y S and
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where
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implies that
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(2) for all x, y, z S and for all
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