Mechanical Engineering Faculty

1. Mechanical Engineering Faculty CONSTRUCTIVE SEMIGROUPS Siniša Crvenković, University of Novi Sad, e-mail: sima@eunet.rs MelanijaMitrović, University of Niš, e-mail: meli@masfak.ni.ac.rs Daniel Abraham Romano East Sarajevo University, e-mail: bato49@hotmail.com Uppsala, 2012

2. CONSTRUCTIVE MATHEMATICS – CM • interpretation of the phrase ”there exists” as ”we can construct” or "we can compute"; • (not only existential quantifier but) allthe logical conectives and quantifiers have to bereinterpreted. ———————– CM ... means mathematics with intuitionistic logic Uppsala, 2012

3. CONSTRUCTIVE MATHEMATICS . . . (in this talk) is Erret Bishop-style mathematics Uppsala, 2012

4. THE (PRE)HISTORY OF INTUITIONISM L. R. J. Brouwer(1881-1966) ( 1.) in classical (traditional) mathemat-ics founded modern topology by establishing • first correct definition of dimension; • topological invariance of dimension; •fixpoint theorem. ( 2.) founded intuitionism • an object only exists after it is con-structed; • he rejects the principle of excluded mid-dle; • actual infinity does not exists, potential infinity does; • no ’sterile’ formalism: only intuitions of the creative subject. Uppsala, 2012

5. HEYTING’S FORMALIZATION ... INTO IL •∀x ∈ AP(x) we have an algorithmthat, applied to an object x and a proof thatx ∈ A, demonstrates that P(x) holds; •∃x(P(x)) means a witness x0such that P(x0) can be computed; •P ∧ Q means that we have both a proof of P and proof of Q • a proof of A ∨ B consists of a proof of A or a proof of B; •¬A means a proof of A is impossible; •A → B means a proof of A can be converted to a proof of B. (Brower-Heyting-Kolmogorov (BHK) interpretation) Uppsala, 2012

6. E. BISHOP’S CM – BISH Three central principles: • every concept is affirmative/positive; • only use relevant definitions; • avoid pseudogeneralities. — E. Bishop: Foundations of Constructive Analysis, McGraw-Hill, New York, 1967. Uppsala, 2012

7. MATHEMATICS IN BISH: some examples Bishop: Every theorem of classical mathe-matics presents a challenge: find a constructive version with a constructive proof. This constructive version can be obtained by strengthening the conditions or weakening the conclusion of the theorem. Uppsala, 2012

8. DOUGLAS S. BRIDGES • there has been a steady stream of publications contributing to Bishops programme since 1967 • one of the most prolific contributor is D. S. Bridges – E. Bishop, D.S. Bridges: Constructive Analysis,GrundlehrendermathematischenWis-senschaften 279, Springer, Berlin, 1985. – D. S. Bridges, F. Richman: Varieties of constructive mathematics, London Mathematical Society Lecture Notes 97, Cambridge University Press, Cambridge, 1987. – D. S. Bridges, L. S. Vita: Techniques in Constructive Analysis,Universitext, Springer, 2006. Uppsala, 2012

9. D. S. Bridges - Constructive Topology • D. S. Bridges and L. S. Vita in the last decade in series of articles have been developed The theory of apartness space, a counterpart of the classical proximity spaces. • NEW - Their systematic research of computable topology using apartness as the fundamental notion, results with the first book with such kind of approach to constructive topology, – D. S. Bridges, L. S. Vita: Apartness and Uniformity - A Constructive Development, CiE series on Theory and Applications of Computability, Springer, 2011. Uppsala, 2012

10. CONSTRUCTIVE ALGEBRA “Contrary to Bishop’s expectations, modern algebra also proved amenable to natural, thor-oughgoing, constructive treatment.” (from – D. S. Bridges, S. Reeves: Constructive Mathematics in Theory and Progamming Practice,PhilosophiaMathematica (3) Vol. 7 (1999) 63-104.) —————— – R. Mines, F. Richman, W. Ruitenburg: A Course of Constructive Algebra; Springer-Verlag, New York 1988. Uppsala, 2012

11. CONSTRUCTIVE ALGEBRA ... is more complicated than classical in various ways • algebraic structure as a rule do not carry a decidable equality relation; • there is (sometime) awkward abundance of all kinds of substructures, and hence of quotient structures. Uppsala, 2012

12. CLASSICAL ALGEBRA - foundational part • the formulation of homomorphic images is one of the principal tools used to manipulate algebras; • concepts of congruence, quotient algebra and homomorphism are closely related; Isomorphism theorems describe the relationship between quotients, homomorphisms and congruences. Uppsala, 2012

13. MAIN TARGET Isomorphism Theorems for Semigroup with Apartness Uppsala, 2012

14. ALGEBRAIC STRUCTURES WITH APARTNESS • A. Heyting (1941) considered structures equipped with an apartness relation in full generality; • B. Jacobs (1995) - algebraic structures with apartness can be applied in computer science (especially in computer programming). • Basic notion: ◦ equality ◦ apartness ◦ order Uppsala, 2012

15. EQUALITY • To define a set (S, =) means that we have ◦ a property that enables us to construct members of S; ◦ described the equality = between elements of S. • S is used to denote a set (S, =). • S is nonempty if we can construct an element of S. • Property P(x) which are extensional in the sense that for all x, x’ ∈ S with x = x′, P(x) and P(x′) are equivalent. Uppsala, 2012

16. SET WITH APARTNESS A binary relation ≠ on S is apartness if it satisfies the axioms of: ¬(x ≠ x) (irreflexivity) x ≠ y ⇒ y ≠ x (symmetry) x ≠ z ⇒ ∀y (x ≠ y ∨ y ≠ z) (cotransitivity) • (S, =,≠) is a set with apartness • tight apartness: ¬(x ≠ y) ⇒ x = y ◦ x ≠ y ∧ y = z ⇒ x ≠ z (by extensionality). Uppsala, 2012

17. MAPPING f : S → T Uppsala, 2012

18. AN IMPORTANT EXAMPLE Uppsala, 2012

19. ISOMORPHISM THEOREMS IN BISH – A.S. Troelstra, D. van Dalen: Constructivism in Mathematics, An Introduction, (two volumes), North - Holland, Amsterdam 1988. • groups with tight apartness normal subgroup — normal antisubgroup • rings with tight apartness ideal — anti-ideal Uppsala, 2012

20. T. S. TROELSTRA, D. van DALEN - GROUPS Uppsala, 2012

21. T. S. TROELSTRA, D. van DALEN - GROUPS Uppsala, 2012

22. T. S. TROELSTRA, D. van DALEN - GROUPS Uppsala, 2012

23. T. S. TROELSTRA, D. van DALEN - RINGS Uppsala, 2012

24. T. S. TROELSTRA, D. van DALEN - RINGS Uppsala, 2012

25. T. S. TROELSTRA, D. van DALEN - RINGS Uppsala, 2012

26. SET WITH APARTNESS A binary relation ≠ on S is apartness if it satisfies the axioms of: ¬(x ≠ x) (irreflexivity) x ≠ y ⇒ y ≠ x (symmetry) x ≠ z ⇒ ∀y (x ≠ y ∨ y ≠ z) (cotransitivity) • (S, =,≠) is a set with apartness Uppsala, 2012

27. COMPLEMENT Uppsala, 2012

28. COMPLEMENT - IMPORTANT EXAMPLE Uppsala, 2012

29. COEQUIVALENCE - EQUIVALENCE Uppsala, 2012

30. COEQUIVALENCE - EQUIVALENCE Uppsala, 2012

31. COFACTOR SET - FACTOR SET Uppsala, 2012

32. APARTNESS ISOMORPHISM THEOREM Uppsala, 2012

33. SEMIGROUPS WITH APARTNESS Uppsala, 2012

34. APARTNESS NEED NOT BE TIGHT Uppsala, 2012

35. COCONGRUENCE - CONGRUENCE Uppsala, 2012

36. COFACTOR - FACTOR SEMIGROUP Uppsala, 2012

37. APARTNESS HOMORPHISM THEOREM Uppsala, 2012

38. D. BRIDGES, F. RICHMAN - “VARIETIES OF CM” Uppsala, 2012

39. Thanks for your attention! Uppsala, 2012

40. Niš2013 Celebrationof the 1700th anniversary of the Edict of Milan, which was signed by emperors Constantine and Licinius in 313 AD and which initiated the era of religious toleration for the Christian faith in the Roman Empire. Constantine ("The Great") was born in the Roman city of Naissus, present-day Niš, in 272 AD. Niš, 2013