Semi-Heyting Algebras and Identities of Associative Type

Juan M. Cornejo; Hanamantagouda P. Sankappanavar

Bulletin of the Section of Logic (2019)

  • Volume: 48, Issue: 2
  • ISSN: 0138-0680

Abstract

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An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras.  They share several important properties with Heyting algebras.  An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of 𝒮ℋ.

How to cite

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Juan M. Cornejo, and Hanamantagouda P. Sankappanavar. "Semi-Heyting Algebras and Identities of Associative Type." Bulletin of the Section of Logic 48.2 (2019): null. <http://eudml.org/doc/295515>.

@article{JuanM2019,
abstract = {An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras.  They share several important properties with Heyting algebras.  An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of 𝒮ℋ.},
author = {Juan M. Cornejo, Hanamantagouda P. Sankappanavar},
journal = {Bulletin of the Section of Logic},
keywords = {semi-Heyting algebra; Heyting algebra; identity of associative type; subvariety of associative type},
language = {eng},
number = {2},
pages = {null},
title = {Semi-Heyting Algebras and Identities of Associative Type},
url = {http://eudml.org/doc/295515},
volume = {48},
year = {2019},
}

TY - JOUR
AU - Juan M. Cornejo
AU - Hanamantagouda P. Sankappanavar
TI - Semi-Heyting Algebras and Identities of Associative Type
JO - Bulletin of the Section of Logic
PY - 2019
VL - 48
IS - 2
SP - null
AB - An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. 𝒮ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras.  They share several important properties with Heyting algebras.  An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of 𝒮ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety 𝒮ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of 𝒮ℋ.
LA - eng
KW - semi-Heyting algebra; Heyting algebra; identity of associative type; subvariety of associative type
UR - http://eudml.org/doc/295515
ER -

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