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The Isomorphism Between Structures Is an Equivalence Relation

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The Isomorphism Between Structures Is an Equivalence Relation Open Journal of Mathematics and Physics | Volume 3, Article 233, ISSN: 2674-5747 https://doi.org/10.31219/osf.io/7uwnh | published: 24 Apr 2021 | https://ojmp.org JD [white paper: pedagogical] Diamond Open Access [waiting peer review] The isomorphism between structures is an equivalence relation Open Mathematics Collaboration∗† September 5, 2021 Abstract We present a pedagogical proof that the function of an isomorphism between two structures is an equivalence relation. keywords: isomorphism, structures, equivalence relation, first-order logic The most updated version of this white paper is available at https://osf.io/7uwnh/download https://zenodo.org/record/5459284 Introduction 1. This is a pedagogical white paper on first-order logic. 2. Our purpose is to discuss a result in [1] which is licensed under [2]. 3. We use minimal notation but preserving all relevant mathematical information. ∗All authors with their affiliations appear at the end of this white paper. †Corresponding author: [email protected] | Open Mathematics Collaboration 1 Meta-linguistic symbols 4. means that what is on the left is defined by what is on the right. 5. ∶= means that the strings on both sides are identical. 6. ∶a≡,b c means deduction of c from a,b. ⊢ Proposition 7. The function of an isomorphism between two structures is an equiva- lence relation. Givens 8. A, B L-structures 9. A B∶= 10. A,$B universes of A and B, respectively 11. Defini!=tion: Two L-structures are isomorphic, written A B, if there exists a bijection ı A B such that $ (a) ı cA cB for!any%constant symbol c of L, (b) ı f A a , ..., a f B ı a , ..., ı a for any n-ary function ( ) =1 n 1 n symbol f of L and any a1, ..., an A, ( ( )) = ( ( ) ( )) (c) a , ..., a RA ı a , ..., ı a RB for any n-ary relation 1 n 1 n& symbol R of L and any a1, ..., an A. ' ( & ) ' ( ) ( )( & & Proof of (7) 12. We need to prove that is an equivalence relation, i.e., (a) A A, (reflexive$) (b) A B B A , (symmetric) $ ( $ ) % ( $ ) 2 (c) A B and B C A C , (transitive). ( $ $ ) % ( $ ) Reflexive 13. First, we prove that A A. 14. Let ı A A such that$ı a a for any a A. 15. It is c!lear%that (14) is a b(ije)ct=ion. & 16. (14) satisfies (11.a). 17. (9), (11.b), (14) ı f a1, ..., an f ı a1 , ..., ı an f a1, ..., an 18. (14) satisfies (11.#c).( ( )) = ( ( ) ( )) = ( ) Symmetric 19. Next, we prove that A B B A . 20. Assume A B. ( $ ) % ( $ ) 21. Due to (20$), there is a bijection ı A B (an isomorphism) such that (11) is satisfied. ! % 22. Let ȷ B A such that ȷ b ı 1 b for any b B. − 23. (21), (!22)% ȷ is a bijectio(n )fr=om B( )into A. & 24. Now we ne#ed to prove that ȷ satisfies (11). 25. We now prove (11.a), i.e., that ȷ cB cA for any constant symbol c of L(. ) = 26. (21), (22) j cB ı 1 cB ı 1 ı cA cA − − # ( ) = ( ) = ( ( )) = 3 27. We now prove (11.b), i.e., that B A ȷ f b1, ..., bn f ȷ b1 , ..., ȷ bn for any n-ary fun(ctio(n symbol)f) =of L(a(nd)any b(1, ..).,)bn B. 28. Hereafter we assume n 1, one can easily generalize the p&roof to n 1. 29. Pick an arbitrary b B=. > 30. Since (21), !a ı a& b. 31. ı is an isom*orph! is(m)f=rom A to B ı f A a f B ı a 32. ȷ f B b ı 1 f B b ı 1 f B ı#a ( (ı )1)ı=f A (a( )) f A a f A ı 1 b − f A ȷ b − − ( ( −)) = ( ( )) = ( ( ( ))) = ( ( ( ))) = ( ) = 33. =We n(ow (pr)o)ve= (11(.c(), )i.)e., that B A b1, ..., bn R ȷ b1 , ..., ȷ bn R for any n-ary r'elation sy(m& bol R)o'f (L )and an(y b)1(,&..., bn A. 34. Again, assume n 1 like in (28). & 35. (11.c), (21) a =RA ı a RB 36. Pick an arbi#trar&y b B)su(ch) &that ı a b. 37. b RB ı a RB& a RA ı (1 b) = RA ȷ b RA − 38. (2&6), (3)2), (37))& i)s a s&ymm)etric(re)la&tion.) ( ) & # $ Transitive 39. Let ı A B be an isomorphism from A to B. 40. Let’s !defi%ne ı A B by ı a b. 41. ı f A a f B! ı %a ( ) = ( ( )) = ( ( )) 4 42. a RA ȷ a RB 43. ⟨Le⟩t ∈ȷ B ↔ C⟨ (be)⟩a∈n isomorphism from B to C. 44. Let’s ∶defin→e ȷ B C by ȷ b c. 45. ȷ f B b f C∶ ȷ b→ ( ) = 46. (b (RB)) = ȷ (b( ))RC 47. ⟨Le⟩t’∈s defi↔ne ⟨k ( A)⟩ ∈ C by k a ȷ ı a . 48. We now prove∶(11→.a), i.e., th(a)t = ( ( )) k cA cC for any constant symbol c of L(. ) = 49. k a ȷ ı a ȷ b c 50. W(e )no=w(p(rov)e) (=11(.b),=i.e., that A C k f a1, ..., an f k a1 , ..., k an for any n-ary fun(ctio(n symbol))f =of L( a(nd)any a1(, ...),)an A. 51. To improve the readability, we will ∈ (a) omit the parenthesis; (b) define (temporarily) A A, B B, and C C. 52. Let’s write in the following ∶w=hat we∶=know from ∶b=oth isomorphisms, (39) and (43), using the notation in (51). 53. ıa b, ȷb c; ȷ 1c b, ı 1b a − − 54. ȷıa= c, ı=1ȷ 1c a= = − − 55. ka =ȷıa , ka =kı 1b kı 1ȷ 1c − − − = = = 5 56. k 1c k 1ȷb k 1ȷıa − − − 57. ıf Aa= f Bıa,= ȷf Bb f Cȷb 58. ȷ 1f C=c f Bȷ 1c, ı =1f Bb f Aı 1b − − − − 59. We nee=d to prove that = kf Aa f Cka. 60. f Cka ȷȷ 1f Cka ȷf Bȷ 1ka ȷıı=1f Bȷ 1ka ȷıf Aı 1ȷ 1ka ȷıf Aa kf Aa − − − − − − = = = = = = 61. We now prove (11.c), i.e., that A C a1, ..., an R k a1 , ..., k an R for any n-ary ⟨relation s⟩ym∈ bol ↔R o⟨f (L a)nd an(y a)1,⟩.∈.., an A. 62. Suppose a RA. ∈ 63. (40), (44), ∈(47) k a ȷ ı a RC 64. Suppose k a ⊢RC.( ) = ( ( )) ∈ 65. Let’s defin(e k) ∈1 C A by k 1 c ı 1 ȷ 1 c . − − − − 66. (39), (40), (43), ∶(44)→, (65) k (1 c) = ı (1 ȷ (1 c)) a RA − − − ⊢ ( ) = ( ( )) = ∈ Final Remarks 67. A B is an equivalence relation ( $ ) ) ($ ) Open Invitation Review, add content, and co-author this white paper [3,4]. Join the Open Mathematics Collaboration. Send your contribution to [email protected]. 6 Open Science The latex file for this white paper together with other supplementary files are available in [5,6]. How to cite this paper? https://doi.org/10.31219/osf.io/7uwnh https://zenodo.org/record/5459284 Acknowledgements Center for Open Science https://cos.io + Open Science Framework https://osf.io + Zenodo https://zenodo.org + Agreement All authors agree with [4]. References [1] Leary, Christopher C., and Lars Kristiansen. A friendly introduction to mathematical logic, 2nd edition, 2015. https://knightscholar.geneseo.edu/geneseo-authors/6 [2] CC. Creative Commons. CC-By Attribution 4.0 International. https://creativecommons.org/licenses/by/4.0 7 [3] Lobo, Matheus P. “Microarticles.” OSF Preprints, 28 Oct. 2019. https://doi.org/10.31219/osf.io/ejrct [4] Lobo, Matheus P. “Simple Guidelines for Authors: Open Journal of Mathematics and Physics.” OSF Preprints, 15 Nov. 2019. https://doi.org/10.31219/osf.io/fk836 [5] Lobo, Matheus P. “Open Journal of Mathematics and Physics (OJMP).” OSF, 21 Apr. 2020. https://osf.io/6hzyp/files [6] https://zenodo.org/record/5459284 The Open Mathematics Collaboration Matheus Pereira Lobo (lead author, [email protected])1,2 https://orcid.org/0000-0003-4554-1372 1Federal University of Tocantins (Brazil) 2Universidade Aberta (UAb, Portugal) 8.
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