Applications of Finite Duality to Locally Finite Varieties of BL-Algebras

Aguzzoli, Stefano; Bova, Simone; Marra, Vincenzo

doi:10.1007/978-3-540-92687-0_1

Cited by 13 publications

(12 citation statements)

References 7 publications

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“…Such relation has been used to establish functorial correspondences before between BL-algebras and certain kind of labeled forests 2 (c.f. [2]). Motivated by these ideas, in this section we show that there exist a functor from the category of finite MTL-algebras to the category of finite labeled forests.…”

Section: Finite Labeled Forestsmentioning

confidence: 99%

On finite MTL-algebras that are representable as poset products of archimedean chains

Castiglioni¹,

Botero²

2020

Fuzzy Sets and Systems

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We obtain a duality between certain category of finite MTL-algebras and the category of finite labeled trees. In addition we prove that certain poset products of MTL-algebras are essentialy sheaves of MTL-chains over Alexandrov spaces. Finally we give a concrete description for the studied poset products in terms of direct products and ordinal sums of finite MTL-algebras.Proof. Let f be an injective morphism of MTL-algebras, then f (x) = 1 = f (1) implies x = 1. On the other hand, let us assume that f (thus by general properties of the residual it follows that f (a) → f (b) = 1 and f (b) → f (a) = 1 so f (a → b) = 1 and f (b → a) = 1. From the assumption we get that a → b = 1 and b → a = 1, thus a ≤ b and b ≤ a. Hence, a = b so f is injective.Lemma 4. Let f : A → B be a morphism of finite MTL-chains. If A is archimedean then B = 1 or f is injective.Proof. Since A is archimedean, by (ii) of Corollary 1 we get that if is also simple so K f = A or K f = {1}. In the first case, we get that f (a) = 1 for every a ∈ A, so in particular f (0) = 0 = 1, hence B = 1. In the last case, it follows that f (a) = 1 implies a = 1, so by Lemma 3 f is injective.

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Section: Finite Labeled Forestsmentioning

confidence: 99%

On finite MTL-algebras that are representable as poset products of archimedean chains

Castiglioni¹,

Botero²

2020

Fuzzy Sets and Systems

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“…In this Section we shall present the relevant facts for our purposes about the dual categorical equivalence between finite BL-algebras (with homomorphisms) and the category of finite weighted forests. The duality is introduced in [4].…”

Section: Finite Bl-algebras and Finite Weighted Forestsmentioning

confidence: 99%

Automorphism Groups of Finite BL-Algebras

Aguzzoli

Gerla

2020

Information Processing and Management of Uncertainty in Knowledge-Based Systems

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“…Given such a mapĥ, the homomorphism h is defined by h(1) = 1 and h Note that the mapĥ is a special case of the weight preserving p-morphisms in [3].…”

Section: Finitely Generated Varieties Of Bl-algebrasmentioning

confidence: 99%