On a Definition of a Variety of Monadic ℓ-Groups

Castiglioni, José Luis; Lewin, Renato A.; Sagastume, Marta

doi:10.1007/s11225-012-9464-1

Cited by 7 publications

(19 citation statements)

References 8 publications

(20 reference statements)

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“…Conversely, if T is an object of DRL in which there exists an operator κ that satisfies the previous equations, then

holds on T [, Theorem 1]. In the following, we use DRL′ to denote the category whose objects have a unary operator κ in its signature satisfying the corresponding equations.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…If

f : A \to B

is a morphism in MV, then

is the morphism in

{MV}^{@ t h •} @ t h •

given by

. The adjunction

is an equivalence [, Corollary 15]. For every

we have the isomorphism

given by

α (a) = (a, \neg a)

, and for every

T \in {MV}^{@ t h •} @ t h •

we have the isomorphism

given by

β (x) = (λ x, λ \sim x)

, where

λ x = \sim κ \sim x

.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…">2.The isomorphism

{MV}^{@ t h •} @ t h •

extends to an isomorphism

. Proof The first isomorphism is given by the assignment

φ (a) = (a, \neg a)

(cf. ). It is easy to see that φ is also a morphism in

{MV}_{U}

.…”

Section: On U‐operators In MV and Mv @Th•@th•mentioning

confidence: 97%

“…Lemma Let

〈T, \land, \lor, *, \to, \sim, κ, 0, 1, c〉 \in {MV}^{@ t h •} @ t h •

and let

x, y \in T

. Then the following conditions hold: (a)

κ 0 = 0

and

κ (c) = κ 1 = 1

, (b)

κ (x \land y) = κ (x) \land κ (y)

, (c)

κ (x \lor y) = κ (x) \lor κ (y)

, (d)

κ (x * y) = (λ x * κ y) \lor (λ y * κ x)

, (e)

κ (x + y) = κ (x) + κ (y)

, where

x + y = \sim ((\sim x) * (\sim y))

, (f)

λ (x * y) = λ x * λ y

, (g)

κ (x \to y) = λ x \to κ y

, (h)

\sim κ x \leq κ \sim x

. Proof It follows from results of . Other possible proof follows from the categorical equivalence between MV and

{MV}^{@ t h •} @ t h •

, and the fact that for

A \in MV

the map

…”

Section: A System Based On the System Boldrwmentioning

confidence: 99%

“…To end the section we extend the categorical equivalence

K ⊣ {sans-serifK}^{@ t h •} @ t h •

(cf. ) by defining two extended functors, that we shall call also

scriptK

and

{sans-serifK}^{@ t h •} @ t h •

, between the categories

{MV}_{U}

, whose objects are pairs formed by an MV‐algebra and a U‐operator, and

, whose objects are pairs formed by an object of

{MV}^{@ t h •} @ t h •

and a

c

U‐operator.…”

Section: On U‐operators In MV and Mv @Th•@th•mentioning

confidence: 99%

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The logic Ł^•

Sagastume

Martín

2014

Math. Log. Quart.

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The algebraic category MV • is the image of MV, the category whose objects are the MV-algebras, by the equivalence K • (cf. [7,8]). In this paper we define the logic Ł • whose Lindenbaum algebra is an MV • -algebra (object of MV • ), and establish a link between Ł • and the infinite valued Łukasiewicz logic Ł. We define cU-operators, that have properties of universal quantifiers, and establish a bijection that maps an MV-algebra endowed with a U-operator (cf. [20][21][22]) into an MV • -algebra endowed with a cU-operator. This map extends to a functor that is a categorical equivalence.

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“…Conversely, if T is an object of DRL in which there exists an operator κ that satisfies the previous equations, then

holds on T [, Theorem 1]. In the following, we use DRL′ to denote the category whose objects have a unary operator κ in its signature satisfying the corresponding equations.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…If

f : A \to B

is a morphism in MV, then

is the morphism in

{MV}^{@ t h •} @ t h •

given by

. The adjunction

is an equivalence [, Corollary 15]. For every

we have the isomorphism

given by

α (a) = (a, \neg a)

, and for every

T \in {MV}^{@ t h •} @ t h •

we have the isomorphism

given by

β (x) = (λ x, λ \sim x)

, where

λ x = \sim κ \sim x

.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…">2.The isomorphism

{MV}^{@ t h •} @ t h •

extends to an isomorphism

. Proof The first isomorphism is given by the assignment

φ (a) = (a, \neg a)

(cf. ). It is easy to see that φ is also a morphism in

{MV}_{U}

.…”

Section: On U‐operators In MV and Mv @Th•@th•mentioning

confidence: 97%

“…Lemma Let

〈T, \land, \lor, *, \to, \sim, κ, 0, 1, c〉 \in {MV}^{@ t h •} @ t h •

and let

x, y \in T

. Then the following conditions hold: (a)

κ 0 = 0

and

κ (c) = κ 1 = 1

, (b)

κ (x \land y) = κ (x) \land κ (y)

, (c)

κ (x \lor y) = κ (x) \lor κ (y)

, (d)

κ (x * y) = (λ x * κ y) \lor (λ y * κ x)

, (e)

κ (x + y) = κ (x) + κ (y)

, where

x + y = \sim ((\sim x) * (\sim y))

, (f)

λ (x * y) = λ x * λ y

, (g)

κ (x \to y) = λ x \to κ y

, (h)

\sim κ x \leq κ \sim x

. Proof It follows from results of . Other possible proof follows from the categorical equivalence between MV and

{MV}^{@ t h •} @ t h •

, and the fact that for

A \in MV

the map

…”

Section: A System Based On the System Boldrwmentioning

confidence: 99%

“…To end the section we extend the categorical equivalence

K ⊣ {sans-serifK}^{@ t h •} @ t h •

(cf. ) by defining two extended functors, that we shall call also

scriptK

and

{sans-serifK}^{@ t h •} @ t h •

, between the categories

{MV}_{U}

, whose objects are pairs formed by an MV‐algebra and a U‐operator, and

, whose objects are pairs formed by an object of

{MV}^{@ t h •} @ t h •

and a

c

U‐operator.…”

Section: On U‐operators In MV and Mv @Th•@th•mentioning

confidence: 99%

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The logic Ł^•

Sagastume

Martín

2014

Math. Log. Quart.

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Kleene algebras with implication

2017

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A Categorical Equivalence Motivated by Kalman’s Construction

Sagastume

Martín

2015

Stud Logica

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An equivalence between the category of MV-algebras and the category MV • is given in Castiglioni et al. (Studia Logica 102(1):67-92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations a = ¬¬a, (a → b) ∨ (b → a) = 1 and a (a → b) = a ∧ b. An object of MV • is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs (A, I), where A is an MV-algebra and I is an ideal of A.

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On a Definition of a Variety of Monadic ℓ-Groups

Cited by 7 publications

References 8 publications

The logic Ł^•

The logic Ł^•

Kleene algebras with implication

A Categorical Equivalence Motivated by Kalman’s Construction

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On a Definition of a Variety of Monadic ℓ-Groups

Cited by 7 publications

References 8 publications

The logic Ł•

The logic Ł•

Kleene algebras with implication

A Categorical Equivalence Motivated by Kalman’s Construction

Contact Info

Product

Resources

About

The logic Ł^•

The logic Ł^•