Norm of formulas: the unit-norm ϕ formula in R L , setting � [ ϕ ] � u = � f ϕ � ∞ . ( Lind R L , n , � · � u ) becomes a normed space. Completion The norm-completion of the normed space ( Lind R L , n , � · � u ) is isometrically isomorphic with ( C ([ ✵ , ✶ ] n ) , � · � ∞ ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 11/40
Norm of formulas: the unit-norm ϕ formula in R L , setting � [ ϕ ] � u = � f ϕ � ∞ . ( Lind R L , n , � · � u ) becomes a normed space. Completion The norm-completion of the normed space ( Lind R L , n , � · � u ) is isometrically isomorphic with ( C ([ ✵ , ✶ ] n ) , � · � ∞ ) . Norm of formulas: the integral norm It is possible to define an integral norm on Lind R L , n . With respect to this norm, the completion of Lind R L , n is a suitable space of integrable function and it is connected to the theory of L -spaces. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 11/40
❑❍❛✉s❞ We now have an appropriate notion of syntactical limit, which is compatible with the semantic notion. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 12/40
We now have an appropriate notion of syntactical limit, which is compatible with the semantic notion. ◮ analyze deductive systems closed to limits, ◮ discuss norm completions in logic, ◮ axiomatize a logic whose models are C ( X ) , for basically disconnected X ∈ ❑❍❛✉s❞ . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 12/40
From Q L to R L Monotone sequences of formulas A sequence ( ϕ n ) n of formulas is 1. increasing if ⊢ ϕ n → ϕ n + ✶ 2. decreasing if ⊢ ϕ n − ✶ → ϕ n . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 13/40
From Q L to R L Monotone sequences of formulas A sequence ( ϕ n ) n of formulas is 1. increasing if ⊢ ϕ n → ϕ n + ✶ 2. decreasing if ⊢ ϕ n − ✶ → ϕ n . Rational approximation For any formula ϕ in R L there exist an increasing sequence of formulas { ψ n } n ∈ N and a decreasing sequence of formulas { χ n } n ∈ N , both in Q L , such that lim n ψ n = ϕ and lim n χ n = ϕ . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 13/40
Deductive systems Clearly, the three logical system we are considering are entangled with each other. Each formula in R L can be approximated by sequences in Q L . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 14/40
Deductive systems Clearly, the three logical system we are considering are entangled with each other. Each formula in R L can be approximated by sequences in Q L . Can be said the same for formulas in Q L wrt formulas in Łukasiewicz logic? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 14/40
Deductive systems Clearly, the three logical system we are considering are entangled with each other. Each formula in R L can be approximated by sequences in Q L . Can be said the same for formulas in Q L wrt formulas in Łukasiewicz logic? If not, how these consideration are reflected on the deductive systems of these logics? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 14/40
Deductive systems Recall that L denotes Łukasiewicz logic. Θ ⊆ Form L , we denote Thm (Θ , L ) = { ϕ ∈ Form L | Θ ⊢ L ϕ } the theory determined by Θ in L . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 15/40
Deductive systems Recall that L denotes Łukasiewicz logic. Θ ⊆ Form L , we denote Thm (Θ , L ) = { ϕ ∈ Form L | Θ ⊢ L ϕ } the theory determined by Θ in L . Analogously for Q L and R L , we get Thm (Θ , Q L ) = { ϕ ∈ Form L | Θ ⊢ Q L ϕ } Thm (Θ , R L ) = { ϕ ∈ Form L | Θ ⊢ R L ϕ } S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 15/40
Ł-generated theories in Q L S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 16/40
Ł-generated theories in Q L It is easy to check that, for any f ∈ DMV n there exist f ∈ MV n such that � f � DMV = � f � DMV . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 16/40
Ł-generated theories in Q L It is easy to check that, for any f ∈ DMV n there exist f ∈ MV n such that � f � DMV = � f � DMV . Thus, via the usual corresponded between filters and deductive systems, Let ϕ be a formula of Q L . There exists a formula β of L such that Thm ( ϕ, Q L ) = Thm ( β, Q L ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 16/40
✶ ✷ ✶ ✷ Ł-generated theories in R L S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 17/40
✶ ✷ Ł-generated theories in R L An ideal I of RMV n , n ∈ N , is said to be norm-closed if, whenever f ✶ , f ✷ , . . . , f m , . . . is a sequence of elements of I and { f m } m ∈ N uniformly converges to f , then f ∈ I . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 17/40
✶ ✷ Ł-generated theories in R L An ideal I of RMV n , n ∈ N , is said to be norm-closed if, whenever f ✶ , f ✷ , . . . , f m , . . . is a sequence of elements of I and { f m } m ∈ N uniformly converges to f , then f ∈ I . For example, any σ -ideal is norm-closed. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 17/40
Ł-generated theories in R L An ideal I of RMV n , n ∈ N , is said to be norm-closed if, whenever f ✶ , f ✷ , . . . , f m , . . . is a sequence of elements of I and { f m } m ∈ N uniformly converges to f , then f ∈ I . For example, any σ -ideal is norm-closed. An infinitary deduction rule ϕ ✶ , ϕ ✷ , . . . , ϕ m , . . . ( ⋆ ) if ϕ = lim m ϕ m then ϕ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 17/40
Ł-generated theories in R L The logic R L ⋆ It is the logic obtained from R L adding the rule ( ⋆ ). S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 18/40
Ł-generated theories in R L The logic R L ⋆ It is the logic obtained from R L adding the rule ( ⋆ ). A consequence The deductive systems of R L ⋆ are in correspondence with norm-closed ideals of the Lindenbaum-Tarki algebra of R L . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 18/40
Ł-generated theories in R L The logic R L ⋆ It is the logic obtained from R L adding the rule ( ⋆ ). A consequence The deductive systems of R L ⋆ are in correspondence with norm-closed ideals of the Lindenbaum-Tarki algebra of R L . Let ϕ be a formula of R L . There exists a sequence of formulas Θ = { ϕ n } n ∈ N ⊆ Form Ł such that Thm ( ϕ, R L ⋆ ) = Thm (Θ , R L ⋆ ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 18/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Di Nola A., Lapenta S., Leuştean I., An infinitary logic for basically disconnected compact Hausdorff spaces , accepted for publication on the Journal of Logic and Computation, arXiv:1709.08397 [math.LO] S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 19/40
Some approaches to ❑❍❛✉s❞ 1. frames of opens → duality with compact regular frames (Isbell) 2. frame of regular opens with a proximity → duality with De Vries algebras (De Vries) S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 20/40
Some approaches to ❑❍❛✉s❞ 1. frames of opens → duality with compact regular frames (Isbell) 2. frame of regular opens with a proximity → duality with De Vries algebras (De Vries) 3. algebras of continuous functions → duality with "norm-complete" lattices of functions (Gelfand, Neumark, Stone, Yosida, Kakutani, Banaschewski) S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 20/40
❑❍❛✉s❞ ❑❍❛✉s❞ ❑❍❛✉s❞ Algebras of continuous functions ◮ By Stone duality, a subcategory of ❑❍❛✉s❞ is equivalent to the category whose objects are element of the finitary variety of Boolean Algebras; S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 21/40
❑❍❛✉s❞ Algebras of continuous functions ◮ By Stone duality, a subcategory of ❑❍❛✉s❞ is equivalent to the category whose objects are element of the finitary variety of Boolean Algebras; ◮ An analogous result is trickier for the whole ❑❍❛✉s❞ : indeed, the dual of ❑❍❛✉s❞ is an infinitary variety (Rosický, Banaschewski, Duskin); S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 21/40
❑❍❛✉s❞ Algebras of continuous functions ◮ By Stone duality, a subcategory of ❑❍❛✉s❞ is equivalent to the category whose objects are element of the finitary variety of Boolean Algebras; ◮ An analogous result is trickier for the whole ❑❍❛✉s❞ : indeed, the dual of ❑❍❛✉s❞ is an infinitary variety (Rosický, Banaschewski, Duskin); ◮ Isbell actually proved that it is "enough" to have a variety in which every function has at most countable arity, and explicitly described this variety; S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 21/40
Algebras of continuous functions ◮ By Stone duality, a subcategory of ❑❍❛✉s❞ is equivalent to the category whose objects are element of the finitary variety of Boolean Algebras; ◮ An analogous result is trickier for the whole ❑❍❛✉s❞ : indeed, the dual of ❑❍❛✉s❞ is an infinitary variety (Rosický, Banaschewski, Duskin); ◮ Isbell actually proved that it is "enough" to have a variety in which every function has at most countable arity, and explicitly described this variety; ◮ Marra and Reggio provided a finite axiomatization for a variety of MV-algebras with an infinitary operation δ : δ -algebras are a finitary variety of infinitary algebras that is dual to ❑❍❛✉s❞ . On C ( X ) , their operator coincides with Isbell’s. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 21/40
❘▼❱ ✵ ✶ ✵ ✶ ✶ How to get compact Hausdorff spaces from Riesz MV-algebras? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 22/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Norm-complete Riesz MV-algebras R ∈ ❘▼❱ semisimple, � · � u : R → [ ✵ , ✶ ] � x � u = min { r ∈ [ ✵ , ✶ ] | x ≤ r ✶ } S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 22/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Norm-complete Riesz MV-algebras R ∈ ❘▼❱ semisimple, � · � u : R → [ ✵ , ✶ ] � x � u = min { r ∈ [ ✵ , ✶ ] | x ≤ r ✶ } A Riesz MV-algebra is norm-complete if it is a complete normed space wrt to � · � u . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 22/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Norm-complete Riesz MV-algebras R ∈ ❘▼❱ semisimple, � · � u : R → [ ✵ , ✶ ] � x � u = min { r ∈ [ ✵ , ✶ ] | x ≤ r ✶ } A Riesz MV-algebra is norm-complete if it is a complete normed space wrt to � · � u . M-spaces An M-space is a Banach lattice (norm-complete Riesz Space) endowed with a norm �·� such that � x ∨ y � = max( � x � , � y � ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 22/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Kakutani’s duality The category of M-spaces and suitable morphisms is dual to the category of compact Hausdorff spaces and continuous maps. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 23/40
How to get compact Hausdorff spaces from Riesz MV-algebras? Kakutani’s duality The category of M-spaces and suitable morphisms is dual to the category of compact Hausdorff spaces and continuous maps. M-spaces and Riesz MV-algebras [A. Di Nola and I. Leuştean, 2014] The category of M-spaces and suitable morphisms is equivalent to the full subcategory of norm-complete Riesz MV-algebras. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 23/40
◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ ▼❱ How to get compact Hausdorff spaces from Riesz MV-algebras? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
◆♦r♠❈♦♠♣❧❡t❡❘▼❱ How to get compact Hausdorff spaces from Riesz MV-algebras? dual M-spaces ❑❍❛✉s❞ dual δ ▼❱ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
How to get compact Hausdorff spaces from Riesz MV-algebras? equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ dual δ ▼❱ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
How to get compact Hausdorff spaces from Riesz MV-algebras? equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ dual δ ▼❱ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
How to get compact Hausdorff spaces from Riesz MV-algebras? equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ dual δ ▼❱ semisimple, complete... can we axiomatize them? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
How to get compact Hausdorff spaces from Riesz MV-algebras? equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ dual δ ▼❱ semisimple, complete... can we axiomatize them? Recalling that the uniform limit of formulas is equivalent to "strong order convergence"... S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 24/40
❘▼❱ ❘▼❱ σ -complete algebras S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 25/40
❘▼❱ σ -complete algebras The category ❘▼❱ σ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 25/40
❘▼❱ σ -complete algebras The category ❘▼❱ σ objects : σ -complete Riesz MV-algebras (i.e. closed to countable suprema), arrows : σ -homomorphisms of Riesz MV-algebras. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 25/40
σ -complete algebras The category ❘▼❱ σ objects : σ -complete Riesz MV-algebras (i.e. closed to countable suprema), arrows : σ -homomorphisms of Riesz MV-algebras. It follows from the general theory of Riesz spaces that: ◮ Any σ -complete Riesz MV-algebra is norm-complete; ◮ for any R ∈ ❘▼❱ σ there exists a basically disconnected compact Hausdorff space X space such that R ≃ C ( X ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 25/40
◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ ❇❉❑❍❛✉s❞ ❘▼❱ What we got: S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 26/40
❇❉❑❍❛✉s❞ ❘▼❱ What we got: equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 26/40
What we got: equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ ❇❉❑❍❛✉s❞ ❘▼❱ σ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 26/40
What we got: equiv dual M-spaces ◆♦r♠❈♦♠♣❧❡t❡❘▼❱ ❑❍❛✉s❞ ❇❉❑❍❛✉s❞ ❘▼❱ σ BDKHausd A compact Hausdorff space is basically disconnected if the closure of any open F σ (i.e. countable union of closed sets) is open. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 26/40
An important remark S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 27/40
An important remark σ -complete Riesz MV-algebras are actually infinitary algebras in the sense of Słomiński. Słomiński J., The theory of abstract algebras with infinitary operations , Instytut Matematyczny Polskiej Akademi Nauk, Warszawa (1959). S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 27/40
An important remark σ -complete Riesz MV-algebras are actually infinitary algebras in the sense of Słomiński. Spoiler: they are an infinitary variery! Słomiński J., The theory of abstract algebras with infinitary operations , Instytut Matematyczny Polskiej Akademi Nauk, Warszawa (1959). S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 27/40
✶ The logic IRL S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 28/40
The logic IRL ◮ Language: the one of R L + � ◮ Axioms: the ones of R L + (S1) ϕ k → � n ∈ N ϕ n , for any k ∈ N ◮ Deduction rules: Modus Ponens + (SUP) ( ϕ ✶ → ψ ) , . . . , ( ϕ k → ψ ) . . . � n ∈ N ϕ n → ψ S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 28/40
The semantics of IRL , main results: ◮ Models of the logic are objects in ❘▼❱ σ , S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 29/40
The semantics of IRL , main results: ◮ Models of the logic are objects in ❘▼❱ σ , ◮ Lind IRL is the smallest σ -complete Riesz MV-algebra that contains Lind R L , S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 29/40
The semantics of IRL , main results: ◮ Models of the logic are objects in ❘▼❱ σ , ◮ Lind IRL is the smallest σ -complete Riesz MV-algebra that contains Lind R L , ◮ alternatively, models are spaces C ( X ) , with X basically disconnected compact Hausdorff space. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 29/40
The semantics of IRL , main results: ◮ Models of the logic are objects in ❘▼❱ σ , ◮ Lind IRL is the smallest σ -complete Riesz MV-algebra that contains Lind R L , ◮ alternatively, models are spaces C ( X ) , with X basically disconnected compact Hausdorff space. Hence, There exists a basically disconnected compact Hausdorff space X such that Lind IRL ≃ C ( X ) . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 29/40
❑❍❛✉s❞ ✵ ✶ ✵ ✶ Functional representations for Lind IRL S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 30/40
✵ ✶ ✵ ✶ Functional representations for Lind IRL On the one end, Lind IRL ≃ C ( X ) , for some basically disconnected X ∈ ❑❍❛✉s❞ . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 30/40
✵ ✶ ✵ ✶ Functional representations for Lind IRL On the one end, Lind IRL ≃ C ( X ) , for some basically disconnected X ∈ ❑❍❛✉s❞ . We tried to get an analogous of the Gleason cover, but for a general space X the construction is very complicated (Jayne, Zakherov and Kuldonov, Vermeer) S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 30/40
Functional representations for Lind IRL On the one end, Lind IRL ≃ C ( X ) , for some basically disconnected X ∈ ❑❍❛✉s❞ . We tried to get an analogous of the Gleason cover, but for a general space X the construction is very complicated (Jayne, Zakherov and Kuldonov, Vermeer) On the other end, we can prove that Lind R L , n ⊆ C ([ ✵ , ✶ ] n ) ⊆ Lind IRL , n ⇒ Lind IRL , n is also isomorphic to some class of non-continuous [ ✵ , ✶ ] n -valued functions! Can we characterize them? S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 30/40
✵ ✶ ✵ ✶ Let’s start with Riesz tribes... S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 31/40
✵ ✶ Let’s start with Riesz tribes... A Riesz tribe over X is a Riesz MV-algebra of [ ✵ , ✶ ] -valued functions over X that are closed under pointwise countable suprema. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 31/40
✵ ✶ Let’s start with Riesz tribes... A Riesz tribe over X is a Riesz MV-algebra of [ ✵ , ✶ ] -valued functions over X that are closed under pointwise countable suprema. The Loomis-Sikorski theorem for Riesz MV-algebras Any σ -complete R Riesz MV-algebra is an homomorphic image of a Riesz tribe T . S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 31/40
Let’s start with Riesz tribes... A Riesz tribe over X is a Riesz MV-algebra of [ ✵ , ✶ ] -valued functions over X that are closed under pointwise countable suprema. The Loomis-Sikorski theorem for Riesz MV-algebras Any σ -complete R Riesz MV-algebra is an homomorphic image of a Riesz tribe T . R = C ( X ) and we say that f ∽ g iff { x ∈ X | f ( x ) � = g ( x ) } is meager. Then R is homomorphic image of: T = { f ∈ [ ✵ , ✶ ] X | there exists g ∈ R : f ∽ g } S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 31/40
✵ ✶ A completeness theorem The class of Dedekind σ -complete Riesz MV-algebras is HSP ([ ✵ , ✶ ]) , the infinitary variety generated by [ ✵ , ✶ ] . by the Loomis-Sikorski theorem. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 32/40
A completeness theorem The class of Dedekind σ -complete Riesz MV-algebras is HSP ([ ✵ , ✶ ]) , the infinitary variety generated by [ ✵ , ✶ ] . by the Loomis-Sikorski theorem. Corollary: IRL is [ ✵ , ✶ ] -complete. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 32/40
❘▼❱ ❘▼❱ ✵ ✶ ✵ ✶ Term functions in σ -complete Riesz MV-algebras S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 33/40
❘▼❱ ✵ ✶ ✵ ✶ Term functions in σ -complete Riesz MV-algebras Absolutely free algebras ◮ Term RMV σ , the set of terms in the language of ❘▼❱ σ , is the absolutely free algebra in the same language, denoted by Term RMV σ ( n ) when only n variables occur. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 33/40
✵ ✶ ✵ ✶ Term functions in σ -complete Riesz MV-algebras Absolutely free algebras ◮ Term RMV σ , the set of terms in the language of ❘▼❱ σ , is the absolutely free algebra in the same language, denoted by Term RMV σ ( n ) when only n variables occur. ◮ for A ∈ ❘▼❱ σ , we get τ : A n → A τ ∈ Term RMV σ ( n ) �→ f A S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 33/40
Term functions in σ -complete Riesz MV-algebras Absolutely free algebras ◮ Term RMV σ , the set of terms in the language of ❘▼❱ σ , is the absolutely free algebra in the same language, denoted by Term RMV σ ( n ) when only n variables occur. ◮ for A ∈ ❘▼❱ σ , we get τ : A n → A τ ∈ Term RMV σ ( n ) �→ f A ◮ RT n = { f τ : [ ✵ , ✶ ] n → [ ✵ , ✶ ] | τ ∈ Term RMV σ ( n ) } is a Riesz tribe. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 33/40
❘▼❱ ❘▼❱ ✶ Free algebras The following hold 1. RT n is the smallest Riesz tribe that contains the projections. S. Lapenta (UNISA) Infinitary logic and DBKHausd-spaces 34/40
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