Abstract
In this study, a term operation Sheffer stroke is presented in a given basic algebra 𝒜 and the properties of the Sheffer stroke reduct of 𝒜 are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.
1 Introduction
Take into consideration the problem of expressing equational concepts as simply as possible with the least number of operations or the least number of axioms and so forth. For an example about the least number of axioms, Tarski solved the related problem for Abelian groups with the single axiom x/(y/(z/(x/y))) = z in terms of the division operation in 1938 [1]. As a typical example, we tackle the use of Sheffer stroke operation in algebraic structures. The Sheffer stroke term operation was firstly given by H. M. Sheffer in 1913 [2]. He proved that all Boolean functions could be transplanted to a single binary operation for term operations. In recent years, the problem for Boolean algebra was solved with a single axiom in terms of the Sheffer Stroke operation [3].
The reduction attempts interest mathematicians who want to use less operations or axioms or formulas for the structures under their considerations. The first implication reduct of Boolean algebras connectives was studied by J. C. Abbott [4] under the name implication algebra. Once the logic of quantum mechanics was axiomatized by means of orthomodular lattices, Abbott obtained implication reducts of orthomodular lattices, called orthoimplication algebras in [5]. Later on, this work was generalized for implication reducts of ortholattices by Chajda and Halaš [6], and by Chajda [7] to orthomodular lattices but without the compatibility condition in Chajda, Halaš and Länger [8].
Basic algebras were introduced in Chajda and Emanovský [9], see also Chajda [10] and Chajda et al. [11] and [12, 13] for further information. Basic algebras are an important concept used in different non-classical logics since they contain orthomodular lattices ℒ = (L; ∨, ∧,⊥, 0, 1), where x ⊕ y = (x ∧ y⊥)∨ y and ¬x = x⊥, and constitute as well as provide an axiomatization of the logic of quantum mechanics along with MV-algebras [14], which get an axiomatization of many-valued Łukasiewicz logics; see Chajda [15] and Chajda et al. [16].
Given that the connective Sheffer stroke operation plays a central role in all mentioned logics above, in general, we would like to characterize this operation in basic algebras.
2 Preliminaries
The following fundamental notions are taken from [17] and [18].
Definition 2.1
A bounded lattice is an algebraic structure ℒ = (L; ∨, ∧, 0, 1) such that ℒ = (L; ∨, ∧) is a lattice having the following properties:
for all x ∈ L, x ∨ 1 = 1 and x ∧ 1 = x,
for all x ∈ L, x ∨ 1 = x and x ∧ 0 = 0.
Definition 2.2
Let ℒ = (L; ∨, ∧) be a lattice. A mapping x ↦ x⊥is called an antitone involution if it satisfies the following:
x ≤ y implies y⊥ ≤ x⊥ (antitone),
x⊥⊥ = x (involution).
Definition 2.3
Let ℒ be a bounded lattice with an antiotone involution. If it satisfies
then x⊥ is the complement of x and ℒ = (L; ∨, ∧, ⊥, 0, 1) is an ortholattice.
Lemma 2.4
Let ℒ = (L; ∨, ∧, ⊥) be a lattice with antitone involution. Then it satisfies the following De Morgan laws:
Definition 2.5
([19]). Let 𝒜 = (A, |) be a groupoid. The operation | is said to be a Sheffer stroke operation if it satisfies the following conditions:
(S1) x|y = y|x,
(S2) (x|x)|(x|y) = x,
(S3) x|((y|z)|(y|z)) = ((x|y)|(x|y))|z,
(S4) (x|((x|x)|(y|y)))|(x|((x|x)|(y|y))) = x.
If additionally it satisfies the identity
(S5) y|(x|(x|x)) = y|y, it is called an ortho Sheffer stroke operation.
Lemma 2.6
([19]). Let 𝒜 = (A, |) be a groupoid. The binary relation ≤ defined on A as below
is an order on A.
Lemma 2.7
([19]). Let | be a Sheffer stroke operation on A and ≤ the induced order of 𝒜 = (A, |). Then
x ≤ y if and only if y|y ≤ x|x,
x|(y|(x|x)) = x|x is the identity of 𝒜,
x ≤ y implies y|z ≤ x|z for all z ∈ A,
a ≤ x and a ≤ y imply x|y ≤ a|a.
Definition 2.8
([20]). A basic algebra is an algebra 𝒜 = (A; ⊕, ¬, 0) of type (2, 1, 0) which satisfies the following axioms:
(BA1) x ⊕ 0 = x,
(BA2) ¬¬x = x,
(BA3) ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x,
(BA4) ¬(¬(¬(¬x ⊕ y) ⊕ y) ⊕ z) ⊕ (x ⊕ z) = ¬0.
As shown in [20], every basic algebra can be thought alternatively as a bounded lattice with section antitone involutions. Now, we are dealing with in the case when for an element x of a basic algebra 𝒜 the negation ¬x is a complement of x in the induced lattice ℒ(𝒜). Then we can give the following lemma and its corollary:
Lemma 2.9
([20]). Let 𝒜 = (A; ⊕, ¬, 0) be a basic algebra and x ∈ A. Then ¬x is a complement of x in the induced lattice ℒ(𝒜) = (A; ∨, ∧, (a)a∈A, 0, 1) if and only if it satisfies x ⊕ x = x.
Corollary 2.10
([20]). If a basic algebra 𝒜 = (A; ⊕, ¬, 0) satisfies the identity x ⊕ x = x, then the induced lattice ℒ(𝒜) is an ortholattice.
3 The Sheffer Stroke Reduction of Basic Algebras
As mentioned in [21], we can consider 𝒜 = (A; ⊕, ¬, 0) alternatively in signature {→, 0}, where x → y = ¬x ⊕ y. When regarding a logic axiomatized by a basic algebra, especially if 𝒜 is an MV-algebra, then it is a many-valued fuzzy logic, or if 𝒜 is an orthomodular lattice, it is a logic of quantum mechanics. Then the operation → plays the role of logic connective implication. There are also some interesting results related to implication operation used (see [22-27]). Starting from this point of view, we construct an alternative signature {|} which consists of only the Sheffer stroke operation. Hence, it is of some importance to investigate the Sheffer stroke reduct in a general setting for basic algebras. Now, we can define the following concept.
Definition 3.1
An algebra (A, |) of type (2) is called a Sheffer stroke basic algebra if the following identities hold:
(SH1) (x|(x|x))|(x|x) = x,
(SH2) (x|(y|y))|(y|y) = (y|(x|x))|(x|x),
(SH3) (((x|(y|y))|(y|y))|(z|z))|((x|(z|z))|(x|(z|z))) = x|(x|x).
First of all, we give some simple properties of Sheffer Stroke basic algebras. Then, we demonstrate that every such algebra has an algebraic constant 1 as is the case for implication basic algebras [20].
Lemma 3.2
Let (A, |) be a Sheffer Stroke basic algebra. Then there exists an algebraic constant element 1 ∈ A and (A, |) meets the following identities:
x|(x|x) = 1,
x|(1|1) = 1,
1|(x|x) = x,
((x|(y|y))|(y|y))|(y|y) = x|(y|y),
(y|(x|(y|y)))|(x|(y|y)) = 1.
Proof
(i) : In (SH3), we substitute [z := y] and [y := x] simultaneously and we carry out (SH1). Then we obtain
When we substitute [x := (x|(y|y))] in the above equation, we derive
Therefore, we get
By using (SH2), we deduce
Thus (A; |) satisfies the identity x|(x|x) = y|(y|y) for all x, y ∈ A. It means that (A; |) contains an algebraic constant which will be denoted by 1 and hence this system satisfies the identity x|(x|x) = 1.
(iii) : Applying (i) in (SH1), we obtain
(υ) : By using (i) in (SH3), we get
Firstly, the substitution [x := y] and [y := x] gives
Substituting [x := (x|(y|y))] and [z := (x|(y|y))] in the latter equation yields
By using (SH3) and the identity (iii), we conclude
(iv) : In (iii), the substitution [x := (x|(y|y))] implies
And using (υ) for the value 1 in the above equation, we get
By using (SH2), we obtain
(ii) : In (υ), if we choose x = y then we get
From the identity (i), we obtain
☐
Theorem 3.3
The axioms of Sheffer Stroke basic algebra are independent.
Proof
To prove this claim, we construct a model for each axiom in which that axiom is false while the others are true. Let 𝒜 = ({0, 1}, |𝒜) be our model defined in the following tables:
(1)Independence of (SH1):
We define the operation |𝒜 in the following table:
Operation Table for Independence of (SH1)
| |𝒜 | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 0 |
Then |𝒜 satisfies (SH2) and (SH3), but not (SH1) since (1|(1|1))|(1|1) = 0 ≠ 1.
(2)Independence of (SH2):
Consider the operation |𝒜, defined as in the following table:
Operation Table for Independence of (SH2)
| |𝒜 | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Then |𝒜 satisfies (SH1) and (SH3), but not (SH2) because if we choose x = 1 and y = 0, then (1|(0|0))|(0|0) = 1 ≠ 0 = (0|(1|1))|(1|1).
(3)Independence of (SH3):
Define the operation |𝒜 as in Table 3:
Operation Table for Independence of (SH3)
| |𝒜 | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
The model |𝒜 satisfies (SH1) and (SH2), but not (SH3). When we choose x = 0 and y = z = 1, we get (((0|(1|1))|(1|1))|(1|1))|((0|(1|1))|(0|(1|1))) = 1 ≠ 0 = 0|(0|0). ☐
To construct a bridge between basic algebras and Sheffer stroke basic algebras, we need the following theorem:
Theorem 3.4
Let 𝒜 = (A; ⊕, ¬, 0) be a basic algebra. We define x|y = ¬x ⊕ ¬y. Then (A; |) is a Sheffer Stroke basic algebra.
Proof
By using (BA1) − (BA4), Lemma 2.9 and Lemma 3.2, we can verify the identities (SH1) − (SH3) as follows:
(SH1)
(SH2)
(SH3)
⃞
To reveal the structure of Sheffer stroke basic algebras, we introduce a partial order relation on A.
Lemma 3.5
Let (A; |) be a Sheffer stroke basic algebra. A binary relation ≤ is defined on A as follows:
Then the binary relation ≤ is a partial order on A such that x ≤ 1 for each x ∈ A. Moreover, we have
for all x, y, z ∈ A.
Proof
• Reflexivity follows from Lemma 3.2 (i).
• Assume that x ≤ y and y ≤ x. Then x|(y|y) = 1 and y|(x|x) = 1. From the hypothesis, (SH2) and Lemma 3.2 (iii), we obtain
• Suppose that x ≤ y and y ≤ z. Then we have x|(y|y) = 1 and y|(z|z) = 1. Using this, (SH3) and Lemma 3.2 (iii), we get
Thus x ≤ z, showing that ≤ is a partial order on A. From Lemma 3.2 (ii), we get x ≤ 1 for each x ∈ A.
Moreover, assume that x ≤ y and z ∈ A. Then
which means y|(z|z) ≤ x|(z|z). Putting here [y := 1], we obtain z = 1|(z|z) ≤ x|(z|z). ☐
The partial order ≤ introduced in the above lemma will be called induced partial order of the Sheffer stroke basic algebra (A; |).
Theorem 3.6
Let (A; |) be a Sheffer stroke basic algebra and ≤ its induced partial order. Then (A; ≤) is a join semi-lattice with the greatest element 1 where x ∨ y = (x|(y|y))|(y|y).
Proof
Using Lemma 3.5 and (SH2), we obtain x ≤ (x|(y|y))|(y|y) and y ≤ (y|(x|x))|(x|x) = (x|(y|y))|(y|y). Hence (x|(y|y))|(y|y) is an upper bound for x and y.
We assume x, y ≤ z. Then by using Lemma 3.5 twice we get
Therefore, (x|(y|y))|(y|y) is the least upper bound for x and y, i.e., x ∨ y = (x|(y|y))|(y|y) is the supremum of x, y. ☐
Let (A; |) be a Sheffer stroke basic algebra. The semilattice (A; ∨) derived in the above theorem will be called the induced semilattice of (A; |).
Theorem 3.7
Let (A; |) be a Sheffer Stroke basic algebra and (A; ∨) its induced semi-lattice. For all p ∈ A, the closed interval [p, 1] is a lattice ([p, 1]; ∨, ∧p,p) with an antitone involution x ↦ xp where
for all x, y ∈ A.
Proof
Let x be in [p, 1]. Assume that x ≤ y. Then x|(y|y) = 1. Now substituting [z := p] into (SH3) we get
Since x|(y|y) = 1, by Lemma 3.2 (iii) we obtain
By the definition of ≤, we get y|(p|p) ≤ x|(p|p), hence yp ≤ xp. So x ↦ xp is a partial order reversing mapping. In (SH3), substitute [y := 1] and [z := p]. Then
By Lemma 3.2 (ii) and (iii) we have
Hence p ≤ x|(p|p), i.e., p ≤ xp. Then x ↦ xp is a mapping of [p, 1] into itself. By Theorem 3.6, xpp = (x|(p|p))|(p|p) = x ∨ p = x and so it is an involution of [p, 1]. Then we can carry out De Morgan laws to show that
This is the infimum of x,y ∈ [p, 1]. Consequently, ([p, 1]; ∨, ∧p,p ) is a lattice with an antitone involution. ☐
Corollary 3.8
Let (A; |) be a Sheffer Stroke basic algebra and ≤ is the induced partial order on this system. Then (A; ≤) is a join-semilattice which has the greatest element 1. For each p ∈ A the closed interval [p, 1] is a basic algebra ([p, 1]; ⊕p, ¬p, p) if x ⊕p y = (x|(p|p))|(y|y) and ¬p x = x|(p|p) are defined for all x, y ∈ A.
From now on, ([p, 1]; ⊕p, ¬p, p) is said to be an interval basic algebra with the greatest element 1 and the least element p. Therefore, Theorem 3.7 corresponds to the semilattice structure of a Sheffer stroke basic algebra. We prove that this explanation is complete, in other words, the other direction of Theorem 3.7 can be obtained.
Theorem 3.9
Let (A; ∨, 1) be a join-semilattice with the greatest element 1 such that for every p ∈ A, the closed interval [p, 1] is a lattice with antitone involution x ↦ xp. If we define x|y = (x ∨ yp)yp, then (A; |) is a Sheffer Stroke basic algebra.
We say that (A; |) is a Sheffer Stroke basic algebra which has the least element if there exists an element 0 ∈ A such that 0 ≤ a for all a ∈ A, where ≤ is the induced partial order. So, any Sheffer Stroke basic algebra with the least element 0 satisfies the identity 0|(x|x) = 1.
The proof of the following theorem is straightforward.
Theorem 3.10
Let (A; |) be a Sheffer Stroke basic algebra which has the least element 0. If we define ¬x = x|(0|0) and x ⊕ y = (x|(0|0))|(y|y), then the system (A; ⊕, ¬, 0) is a basic algebra and x|y = ¬x ⊕ ¬y.
In the remaining part of this work, we show that there is a bridge between Sheffer stroke basic algebras and Boolean algebras.
Lemma 3.11
Let (A; |) be a Sheffer stroke basic algebra and (A; ∨) its induced complemented semilattice. For each p ∈ A, the closed interval [p, 1] is a lattice ([p, 1]; ∨, ∧p ,p ) with an antitone involution. Then the following identities hold
for each x ∈ [p, 1].
Proof
The definition of Sheffer stroke operation gives
Substituting [y := (p|p)] into the latter equation yields
From the definition of ≤ we have
From (SH2) we get
So we can conclude that x ∨ (x|(p|p)) = 1 and x ∨ xp = 1. Now we have
Since p is the least element, we have x = x ∨ p = (x|(p|p))|(p|p). We use this equality in the latter equation and by Lemma 3.2 (iii) we obtain
☐
Corollary 3.12
Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1. Then (A; ∨, ∧0,0, 0, 1) is a lattice with an antitone involution.
Lemma 3.13
Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0,0, 0, 1) is a lattice with an antitone involution x ↦ x0. Then it satisfies the following properties for all x ∈ A:
x|(0|0) = 0 if and only if x = 1,
x|(k|k) = k if and only if x ∨ k = 1,
0|0 = 1 ,
x|1 = x|x,
x0 = x|x,
(k|k)|(x|x) = 1 if and only if x ∨ k = 1.
Proof
(⇒:) Assume that x|(0|0) = 0. Then (x|(0|0))|(0|0) = 1. Since x ↦ x0 is an involution, we have 1 = (x|(0|0))|(0|0) = x00 = x.
(⇐:) It follows from Lemma 3.2 (iii).
Assume that x ∈ A. Then 0 ≤ x and 0 ≤ (x|x). By Lemma 2.7 (iv), we obtain x|(x|x) ≤ 0|0 ≤ 1, hence 0|0 = 1.
It follows from Lemma 2.6.
By (iv), we have x0 = x|(0|0) = x|1 = x|x.
(⇒:) In [28], we substitute [x := (k|k)] and [y := x] in the equation x|(y|y) = x|(x|y) to
(⇐:) It is verified similarly. ☐
Theorem 3.14
Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0, 0, 0, 1) its induced complemented lattice with an antitone involution x ↦ x0. Then, there exists unique x0 such that
for all x ∈ A.
Proof
We show that there exists x ↦ x0 an antitone involution of A such that x0 ∨ x = 1 and x0 ∧0 x = 0 in Lemma 3.11. For the uniqueness, assume that x0 = k and x0 = l. Then by Lemma 3.13 (vi), we have
Then from these equalities we get
Using the same technique, we can obtain k ≤ l. Therefore, k = l. Hence, we have unique x0 such that x0 ∨ x = 1 for each x ∈ A. The identity x ∧0 x0 = 0 is verified similarly. ☐
Definition 3.15.
A Sheffer Stroke basic algebra (A; |) is commutative if it satisfies
for all x, y ∈ [p, 1].
Lemma 3.16
Let (A; |) be a commutative basic algebra. Then the interval basic algebra ([p, 1]; ⊕p, ¬p, p) is commutative for each p ∈ A.
From Theorem 2.8 in [20] we obtain the following corollary:
Corollary 3.17
Let (A; |) be a commutative Sheffer Stroke basic algebra and (A; ∨) its induced semilattice. Then
the interval basic algebra ([p, 1]; ⊕p, ¬p, p) is commutative basic algebra for each p ∈ A,
the interval lattice ([p, 1], ∨, ∧p) is distributive for each p ∈ A.
Theorem 3.18
Let (A; |) be a Sheffer Stroke basic algebra with the least element 0 and the greatest element 1, and (A; ∨, ∧0,0, 0, 1) its induced lattice with an antitone involution x ↦ x0. Then (A; ∨, ∧0,0, 0, 1) is Boolean algebra.
Proof
It follows fromTheorem 3.10 and Corollary 2.10. ☐
Acknowledgement
The authors thank the anonymous referees for his/her remarks which helped him to improve the presentation of the paper.
References
[1] Tarski A., Ein beitrag zur axiomatik der abelschen gruppen, Fundamenta Mathematicae, 1938, 30, 253-25610.4064/fm-30-1-253-256Search in Google Scholar
[2] Sheffer H.M., A set of five independent postulates for Boolean algebras, with application to logical constants, Transactions of the American Mathematical Society, 1913, 14, 481-48810.1090/S0002-9947-1913-1500960-1Search in Google Scholar
[3] McCune W., Verof R., Fitelson B., Harris K., Feist A., Wos L., Short single axioms for Boolean algebra, Journal of Automated Reasoning, 2002, 29, 1-1610.1023/A:1020542009983Search in Google Scholar
[4] Abbott J.C., Semi-boolean algebra, Matematički Vesnik, 1967, 40, 177-198Search in Google Scholar
[5] Abbott J.C., Implicational algebras, Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, 1967, 11,3-23Search in Google Scholar
[6] Chajda I., Halaš R., An implication in orthologic, International Journal of Theoretical Physics, 2005, 44, 735-74410.1007/s10773-005-7051-1Search in Google Scholar
[7] Chajda I., Orthomodular semilattices, Discrete mathematics, 2007, 307, 115-11810.1016/j.disc.2006.05.040Search in Google Scholar
[8] Chajda I., Halaš R., Länger H., Orthomodular implication algebras, International Journal of Theoretical Physics, 2001, 40, 1875-188410.1023/A:1011933018776Search in Google Scholar
[9] Chajda I., Emanovský P, Bounded lattices with antitone involutions and properties of MV-algebras, Discussiones Mathematicae, General Algebra and Applications, 2004, 24, 32-4210.7151/dmgaa.1073Search in Google Scholar
[10] Chajda I., Lattices and semilattices having an antitone involution in every upper interval, Comment. Math. Univ. Carolin, 2003, 44, 577-585Search in Google Scholar
[11] Chajda I., Kühr J., Basic algebras, Clone Theory and Discrete Mathematics Algebra and Logic Related to Computer Science, 2013Search in Google Scholar
[12] Chajda I., Kolařík M., Independence of axiom system of basic algebras, Soft Computing, 2009, 13, 41-4310.1007/s00500-008-0291-2Search in Google Scholar
[13] Chajda I., Kolařík M., Interval basic algebras, Novi Sad J. Math, 2009, 39, 71-78Search in Google Scholar
[14] Cignoli R. L., d’Ottaviano I. M. , Mundici D., Algebraic foundations of many-valued reasoning, Springer Science and Business Media, 201310.1007/978-94-015-9480-6Search in Google Scholar
[15] Chajda I., Basic algebras and their applications, an overview, Proceedings of the Salzburg Conference, Verlag Johannes Heyn, Klagenfurt, 2011Search in Google Scholar
[16] Chajda I., Halaš R., Kühr J., Many-valued quantum algebras, Algebra Universalis, 2009, 60, 63-9010.1007/s00012-008-2086-9Search in Google Scholar
[17] Birkhoff G., Lattice Theory, Proc. Amer. Math. Soc., Providence, R. I. third edition, 196710.1090/coll/025Search in Google Scholar
[18] Grätzer G., General Lattice Theory, Springer Science and Business Media, 2002Search in Google Scholar
[19] Chajda I., Sheffer operation in ortholattices, Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium Mathematica, 2005, 44, 19-23Search in Google Scholar
[20] Chajda I., Basic algebras, logics, trends and applications, Asian-European Journal of Mathematics, 2015, 8, (46 pages)10.1142/S1793557115500400Search in Google Scholar
[21] Chajda I., Kolařík M., Švrček F., Implication and equivalential reducts of basic algebras, Acta Universitatis Palackianae Olomucensis Facultas Rerum Naturalium Mathematica, 2010, 49, 21-36Search in Google Scholar
[22] Chajda I., Halaš R., Länger H., Simple axioms for orthomodular implication algebras, International Journal of Theoretical Physics, 2004, 43, 911-91410.1023/B:IJTP.0000048587.50827.93Search in Google Scholar
[23] Chajda I., Filters of implication reducts of basic algebras, Soft Computing, 2014, 18, 849-85210.1007/s00500-013-1200-xSearch in Google Scholar
[24] Chajda I., Halaš R., Kühr J., Semilattice structures, Heldermann, Michigan University, 2007Search in Google Scholar
[25] Piziak R., Orthomodular lattices as implication algebras, Journal of Philosophical Logic, 1974, 3, 413-41810.1007/BF00257483Search in Google Scholar
[26] Pavicic M., Megill N. D., Quantum and classical implication algebras with primitive implications, International Journal of Theoretical Physics, 1998, 37, 2091-209810.1023/A:1026685801864Search in Google Scholar
[27] Herman L., Marsden E. L., Piziak R., Implication connectives in orthomodular lattices, Notre Dame Journal of Formal Logic, 1975, 16, 305-32810.1305/ndjfl/1093891789Search in Google Scholar
[28] Łukaszuk A., Grabowski A., Short Sheffer Stroke-Based Single Axiom for Boolean Algebras, Formalized Mathematics, 1975, 12, 363-370Search in Google Scholar
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