Mechanical Engineering Faculty

CONSTRUCTIVE SEMIGROUPS

Siniša Crvenković,University of Novi Sad, e-mail: sima@eunet.rsMelanija Mitrović,

University of Niš, e-mail: meli@masfak.ni.ac.rsDaniel Abraham Romano

East Sarajevo University, e-mail: bato49@hotmail.com

Uppsala, 2012

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) all the logical conectives and quantifiers have to be reinterpreted.———————–CM... means mathematics with intuitionistic logic

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CONSTRUCTIVE MATHEMATICS

. . . (in this talk) is

Erret Bishop-style mathematics

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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.

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A. HEYTING’S FORMALIZATION ... INTO IL

• ∀x AP(x) ∈ we have an algorithm that, applied to an object x and a proof that x A, ∈demonstrates that P(x) holds;

• ∃x(P(x)) means a witness x0 such 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)

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E. BISHOP’S CM – BISH

Three central principles:

• every concept is affirmative/positive; • only use relevant definitions; • avoid pseudogeneralities.

— E. Bishop: Foundations of ConstructiveAnalysis, McGraw-Hill, New York, 1967.

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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 bystrengthening the conditions or weakening theconclusion of the theorem.

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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, Grundlehren der mathematischen Wis-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.

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D. S. Bridges - Constructive Topology

• D. S. Bridges and L. S. Vita in the lastdecade 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 andUniformity - A Constructive Development,CiE series on Theory and Applications of Computability,

Springer, 2011.

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CONSTRUCTIVE ALGEBRA

“Contrary to Bishop’s expectations, modernalgebra also proved amenable to natural, thor-oughgoing, constructive treatment.”

(from – D. S. Bridges, S. Reeves: Constructive Mathematics in Theory and ProgammingPractice, Philosophia Mathematica (3) Vol. 7(1999) 63-104.)

——————– R. Mines, F. Richman, W. Ruitenburg:A Course of Constructive Algebra; Springer-Verlag, New York 1988.

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CONSTRUCTIVE ALGEBRA

... is more complicated than classical invarious ways

• algebraic structure as a rule do notcarry a decidable equality relation;

• there is (sometime) awkward abundanceof all kinds of substructures, and hence of quotient structures.

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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, homomorphismsand congruences.

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MAIN TARGET

Isomorphism Theorems

for

Semigroup with Apartness

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ALGEBRAIC STRUCTURES WITH APARTNESS

• A. Heyting (1941) considered structuresequipped with an apartness relation in full generality;

• B. Jacobs (1995) - algebraic structures withapartness can be applied in computer science(especially in computer programming).

• Basic notion:◦ equality◦ apartness◦ order

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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.∈

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SET WITH APARTNESS

A binary relation ≠ on S is apartness if itsatisfies 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).∧ ⇒

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MAPPING f : S → T

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AN IMPORTANT EXAMPLE

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ISOMORPHISM THEOREMS IN BISH

– A.S. Troelstra, D. van Dalen: Constructivism in Mathematics, An Introduction, (two volumes), North - Holland, Amsterdam 1988.

• groups with tight apartnessnormal subgroup — normal antisubgroup

• rings with tight apartnessideal — anti-ideal

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T. S. TROELSTRA, D. van DALEN - GROUPS

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T. S. TROELSTRA, D. van DALEN - GROUPS

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T. S. TROELSTRA, D. van DALEN - GROUPS

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T. S. TROELSTRA, D. van DALEN - RINGS

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T. S. TROELSTRA, D. van DALEN - RINGS

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T. S. TROELSTRA, D. van DALEN - RINGS

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SET WITH APARTNESS

A binary relation ≠ on S is apartness if itsatisfies 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

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COMPLEMENT

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COMPLEMENT - IMPORTANT EXAMPLE

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COEQUIVALENCE - EQUIVALENCE

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COEQUIVALENCE - EQUIVALENCE

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COFACTOR SET - FACTOR SET

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APARTNESS ISOMORPHISM THEOREM

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SEMIGROUPS WITH APARTNESS

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APARTNESS NEED NOT BE TIGHT

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COCONGRUENCE - CONGRUENCE

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COFACTOR - FACTOR SEMIGROUP

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APARTNESS HOMORPHISM THEOREM

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

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Thanks for your attention!

Niš, 2013 40

Niš 2013

Celebration of 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.