2012
DOI: 10.1007/s11083-012-9266-0
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Semi-Heyting Algebras Term-equivalent to Gödel Algebras

Abstract: In this paper we investigate those subvarieties of the variety SH of semi-Heyting algebras which are term-equivalent to the variety L H of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety L Com of commutative semi-Heyting algebras and the variety L ∨ generated by the chains in which a < b implies a → b = b . We also study the variety C generated within SH by L H , L ∨ and L Com . In particular we prove that C is locally finite and we obtain … Show more

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Cited by 14 publications
(10 citation statements)
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“…Recall that every semi-Heyting algebra A, ∨, ∧, →, 0, 1 gives rise naturally to a Heyting algebra A, ∨, ∧, → H , 0, 1 by defining the implication x → H y as x → (x ∧ y) (see [ACDV13]). Proof Let A ∈ DHMSH, Γ ∪ {φ} ⊆ Fm and h ∈ Hom(Fm L , A), such that Γ ⊢ DMSH φ and hΓ ⊆ {⊤}.…”
Section: Completeness Of Dhmshmentioning
confidence: 99%
“…Recall that every semi-Heyting algebra A, ∨, ∧, →, 0, 1 gives rise naturally to a Heyting algebra A, ∨, ∧, → H , 0, 1 by defining the implication x → H y as x → (x ∧ y) (see [ACDV13]). Proof Let A ∈ DHMSH, Γ ∪ {φ} ⊆ Fm and h ∈ Hom(Fm L , A), such that Γ ⊢ DMSH φ and hΓ ⊆ {⊤}.…”
Section: Completeness Of Dhmshmentioning
confidence: 99%
“…There exists already some literature related to this problem. The papers that deal with this problem algebraically include [38], [2], [3], [4], [5], [15] and [17]. The paper [4] investigates the properties of semi-Heyting chains and the structure of the variety CSH generated by all semi-Heyting chains.…”
Section: Conjecturementioning
confidence: 99%
“…In [2], it is proved, among other things, that the variety of Boolean semi-Heyting algebras (algebras with an underlying structure of Boolean lattice) constitutes a reflective subcategory of SH, extending the corresponding result for Heyting algebras (see [6,Corollary IX.5.4], and that the free algebras in a subvariety V of SH are directly indecomposable if and only if V satisfies the Stone identity, extending a known result for Heyting algebras. Article [3] presents two other subvarieties of semi-Heyting algebras that are term-equivalent to the variety of Goedel algebras (linear Heyting algebras), and that they are the only other subvarieties in L with this property. The variety of semi-Nelson algebras is introduced in [17] so that the well-known and well-exploited relationship between Heyting and Nelson algebras extends to semi-Heyting and semi-Nelson algebras.…”
Section: Conjecturementioning
confidence: 99%
“…Semi-Heyting algebras share with Heyting algebras the following properties: they are pseudocomplemented and their congruences are determined by the lattice filters. The relationship between the variety of semi-Heyting algebras and the varieties of Heyting algebras (and its expansions) has been also studied in [1][2][3][4]19]. In [21] it was proved that Heyting algebras are semi-Heyting algebras, which satisfy the equation (x ∧ y) → x = 1.…”
Section: Preliminariesmentioning
confidence: 99%