A note on the Hilbert algebras with infimum

Figallo, Aldo V.; Saad, Susana Marta Isay

doi:10.21711/231766362003/rmc242

Cited by 10 publications

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“…In this section we recall definitions and properties about the algebras we will consider later: hemi-implicative semilattices (lattices) [21], Hilbert algebras with infimum [12], implicative semilattices [17] and semi-Heyting algebras [20]. A bounded hemi-implicative semilattice is an algebra (H, ∧, →, 0, 1) of type (2, 2, 0, 0) such that (H, ∧, →, 1) is a hemi-implicative semilattice and 0 is the first element with respect to the order.…”

Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

“…In [12] it is proved that the class of Hilbert algebras with infimum is a variety. We note that this result also follows from the results given by P. M. Idziak in [14] for BCK-algebras with lattice operations.…”

Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

“…Remark 31. A moment of reflection shows that implicative semilattices are Hilbert algebras with infimum where the implication is the right residuum of the infimum, or equivalently, where the following equation holds [12]: a ≤ b → (a∧b). Alternatively, it follows from Lemma 30 that an implicative semilattice is a hemi-implicative semilattice which satisfies a → (b…”

Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

“…A hemi-implicative lattice is an algebra (H, ∧, ∨, →, 0, 1) of type (2, 2, 2, 0, 0) such that (H, ∧, ∨, 0, 1) is a bounded distributive lattice and the reduct algebra (H, ∧, →, 1) is a hemi-implicative semilattice. Implicative semilattices [17] and Hilbert algebras with infimum [12] are examples of hemi-implicative semilattices. Semi-Heyting algebras [20] and some algebras studied in [5] are examples of hemi-implicative lattices.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we apply the mentioned equivalence in order to build up an equivalence for the category whose objects are bounded semilattices and whose morphisms are the corresponding algebra homomorphisms. In Section 4 we recall definitions and properties about hemi-implicative semilattices (lattices) [21], Hilbert algebras with infimum [12], implicative semilattices [17] and semi-Heyting algebras [20]. In Section 5 we employ results of sections 3 and 4 in order to establish equivalences, following the original Kalman's construction, for the categories of bounded hemi-implicative semilattices, bounded Hilbert algebras with infimum, bounded implicative semilattices, hemi-implicative lattices, respectively, and the category of semi-Heyting algebras.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices

Jansana

Martín

2017

Logic Journal of the IGPL

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Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present paper. A hemi-implicative semilattice is an algebra (H, ∧, →, 1) of type (2, 2, 0) such that (H, ∧) is a meet semilattice, 1 is the greatest element with respect to the order, a → a = 1 for every a ∈ H and for every a, b, c ∈ H, if a ≤ b → c then a ∧ b ≤ c. A bounded hemiimplicative semilattice is an algebra (H, ∧, →, 0, 1) of type (2, 2, 0, 0) such that (H, ∧, →, 1) is a hemi-implicative semilattice and 0 is the first element with respect to the order. A hemi-implicative lattice is an algebra (H, ∧, ∨, →, 0, 1) of type (2, 2, 2, 0, 0) such that (H, ∧, ∨, 0, 1) is a bounded distributive lattice and the reduct algebra (H, ∧, →, 1) is a hemi-implicative semilattice.In this paper we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.

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Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

Section: The Variety Of Hemi-implicative Semilattices (Lattices)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices

Jansana

Martín

2017

Logic Journal of the IGPL

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Spectral-like duality for distributive Hilbert algebras with infimum

Celani

Esteban

2017

Algebra Univers.

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Distributive Hilbert algebras with infimum, or DH ∧ -algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH ∧ -algebras and ∧-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH ∧ -algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH ∧ -algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.

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l-Hemi-Implicative Semilattices

Castiglioni

Martín

2017

Stud Logica

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An l-hemi-implicative semilattice is an algebra A = (A, ∧, →, 1) such that (A, ∧, 1) is a semilattice with a greatest element 1 and satisfies: (1) for every a, b, c ∈ A, a ≤ b → c implies a ∧ b ≤ c and (2) a → a = 1. An l-hemi-implicative semilattice is commutative if if it satisfies that a → b = b → a for every a, b ∈ A. It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemiimplicative semilattice it is possible to define an derived operation by a ∼ b := (a → b) ∧ (b → a). Endowing (A, ∧, 1) with the binary operation ∼ the algebra (A, ∧, ∼, 1) results an l-hemi-implicative semilattice, which also satisfies the identity a ∼ b = b ∼ a.In this article, we characterize the (derived) commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element (possibly with bottom). Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices.

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A note on the Hilbert algebras with infimum

Cited by 10 publications

References 0 publications

On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices

On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices

Spectral-like duality for distributive Hilbert algebras with infimum

l-Hemi-Implicative Semilattices

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