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Stone Duality and Model Theory Stone duality and model theory Sam v. Gool and Benjamin Steinberg University of Amsterdam and City College of New York October 8, 2016 AMS Fall Western Sectional Meeting Special Session on Algebraic Logic University of Denver v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 1 / 14 Such monoids are Stone dual spaces of certain Boolean algebras. These Boolean algebras, in turn, are Lindenbaum algebras of certain logical theories. How can we use techniques from logic to study profinite monoids? Which of these techniques are useful more generally, in other (profinite) contexts? Introduction In the theory of regular languages of finite words, profinite monoids are useful canonical objects. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 2 / 14 These Boolean algebras, in turn, are Lindenbaum algebras of certain logical theories. How can we use techniques from logic to study profinite monoids? Which of these techniques are useful more generally, in other (profinite) contexts? Introduction In the theory of regular languages of finite words, profinite monoids are useful canonical objects. Such monoids are Stone dual spaces of certain Boolean algebras. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 2 / 14 How can we use techniques from logic to study profinite monoids? Which of these techniques are useful more generally, in other (profinite) contexts? Introduction In the theory of regular languages of finite words, profinite monoids are useful canonical objects. Such monoids are Stone dual spaces of certain Boolean algebras. These Boolean algebras, in turn, are Lindenbaum algebras of certain logical theories. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 2 / 14 Which of these techniques are useful more generally, in other (profinite) contexts? Introduction In the theory of regular languages of finite words, profinite monoids are useful canonical objects. Such monoids are Stone dual spaces of certain Boolean algebras. These Boolean algebras, in turn, are Lindenbaum algebras of certain logical theories. How can we use techniques from logic to study profinite monoids? v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 2 / 14 Introduction In the theory of regular languages of finite words, profinite monoids are useful canonical objects. Such monoids are Stone dual spaces of certain Boolean algebras. These Boolean algebras, in turn, are Lindenbaum algebras of certain logical theories. How can we use techniques from logic to study profinite monoids? Which of these techniques are useful more generally, in other (profinite) contexts? v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 2 / 14 For example, the finite fa; bg-word abbaa is (f1; 2; 3; 4; 5g; <; Pa; Pb) with Pa = f1; 4; 5g and Pb = f2; 3g. The first-order sentence 9x9y(x < y ^ Pa(x) ^ Pb(x)) is true in this word, but false in, e.g., bba. A Any first-order sentence φ in the signature L = f<g [ fPa j a 2 Ag defines the language Lφ of finite A-words in which φ is true, i.e., Lφ := Modfin(φ). First-order definable languages When defining languages with logic, a finite A-word is seen as a structure with a binary relation < and unary letter predicates (Pa)a2A. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 3 / 14 A Any first-order sentence φ in the signature L = f<g [ fPa j a 2 Ag defines the language Lφ of finite A-words in which φ is true, i.e., Lφ := Modfin(φ). First-order definable languages When defining languages with logic, a finite A-word is seen as a structure with a binary relation < and unary letter predicates (Pa)a2A. For example, the finite fa; bg-word abbaa is (f1; 2; 3; 4; 5g; <; Pa; Pb) with Pa = f1; 4; 5g and Pb = f2; 3g. The first-order sentence 9x9y(x < y ^ Pa(x) ^ Pb(x)) is true in this word, but false in, e.g., bba. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 3 / 14 First-order definable languages When defining languages with logic, a finite A-word is seen as a structure with a binary relation < and unary letter predicates (Pa)a2A. For example, the finite fa; bg-word abbaa is (f1; 2; 3; 4; 5g; <; Pa; Pb) with Pa = f1; 4; 5g and Pb = f2; 3g. The first-order sentence 9x9y(x < y ^ Pa(x) ^ Pb(x)) is true in this word, but false in, e.g., bba. A Any first-order sentence φ in the signature L = f<g [ fPa j a 2 Ag defines the language Lφ of finite A-words in which φ is true, i.e., Lφ := Modfin(φ). v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 3 / 14 Here, a language L ⊆ A∗ is recognizable by a finite monoid M if there is a homomorphism η : A∗ ! M with L = η−1η(L), a monoid is aperiodic if it has no non-trivial subgroups. The class of finite aperiodic monoids is denoted by A. Observation The assignment φ 7! Lφ defines an isomorphism between the A Lindenbaum algebra L(Tfin) of the first-order theory of finite A-words and the Boolean algebra RecA(A) of A-recognizable languages. First-order definability and aperiodicity Theorem (McNaughton & Papert 1971, Sch¨utzenberger 1965) A language is first-order definable if, and only if, it is recognizable by a finite aperiodic monoid. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 4 / 14 a monoid is aperiodic if it has no non-trivial subgroups. The class of finite aperiodic monoids is denoted by A. Observation The assignment φ 7! Lφ defines an isomorphism between the A Lindenbaum algebra L(Tfin) of the first-order theory of finite A-words and the Boolean algebra RecA(A) of A-recognizable languages. First-order definability and aperiodicity Theorem (McNaughton & Papert 1971, Sch¨utzenberger 1965) A language is first-order definable if, and only if, it is recognizable by a finite aperiodic monoid. Here, a language L ⊆ A∗ is recognizable by a finite monoid M if there is a homomorphism η : A∗ ! M with L = η−1η(L), v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 4 / 14 The class of finite aperiodic monoids is denoted by A. Observation The assignment φ 7! Lφ defines an isomorphism between the A Lindenbaum algebra L(Tfin) of the first-order theory of finite A-words and the Boolean algebra RecA(A) of A-recognizable languages. First-order definability and aperiodicity Theorem (McNaughton & Papert 1971, Sch¨utzenberger 1965) A language is first-order definable if, and only if, it is recognizable by a finite aperiodic monoid. Here, a language L ⊆ A∗ is recognizable by a finite monoid M if there is a homomorphism η : A∗ ! M with L = η−1η(L), a monoid is aperiodic if it has no non-trivial subgroups. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 4 / 14 Observation The assignment φ 7! Lφ defines an isomorphism between the A Lindenbaum algebra L(Tfin) of the first-order theory of finite A-words and the Boolean algebra RecA(A) of A-recognizable languages. First-order definability and aperiodicity Theorem (McNaughton & Papert 1971, Sch¨utzenberger 1965) A language is first-order definable if, and only if, it is recognizable by a finite aperiodic monoid. Here, a language L ⊆ A∗ is recognizable by a finite monoid M if there is a homomorphism η : A∗ ! M with L = η−1η(L), a monoid is aperiodic if it has no non-trivial subgroups. The class of finite aperiodic monoids is denoted by A. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 4 / 14 First-order definability and aperiodicity Theorem (McNaughton & Papert 1971, Sch¨utzenberger 1965) A language is first-order definable if, and only if, it is recognizable by a finite aperiodic monoid. Here, a language L ⊆ A∗ is recognizable by a finite monoid M if there is a homomorphism η : A∗ ! M with L = η−1η(L), a monoid is aperiodic if it has no non-trivial subgroups. The class of finite aperiodic monoids is denoted by A. Observation The assignment φ 7! Lφ defines an isomorphism between the A Lindenbaum algebra L(Tfin) of the first-order theory of finite A-words and the Boolean algebra RecA(A) of A-recognizable languages. v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 4 / 14 Alternatively, a profinite monoid is a projective limit of finite discrete monoids in the category of topological monoids. The free profinite monoid over a finite set A, Ac∗, has as its clopen sets the closures of languages L over A that are recognizable by a finite monoid. Profinite monoids A profinite monoid is a topological monoid whose underlying space is Boolean (compact, T2, 0-dim). v. Gool and Steinberg (UvA & CCNY) Stone duality and model theory AMS Denver, Alg. Log. 5 / 14 The free profinite monoid over a finite set A, Ac∗, has as its clopen sets the closures of languages L over A that are recognizable by a finite monoid. Profinite monoids A profinite monoid is a topological monoid whose underlying space is Boolean (compact, T2, 0-dim). Alternatively, a profinite monoid is a projective limit of finite discrete monoids in the category of topological monoids.
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