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<p>The variety of weak Heyting algebras corresponds to the strict implication fragment of the normal modal logic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{K}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>K</mtext> </math></EquationSource> </InlineEquation>, which is also known as the subintuitionistic local consequence of the class of all Kripke models.</p><p>In this paper we study the variety of commutative weak Heyting algebras, which is defined as the subvariety of weak Heyting algebras whose members satisfy the identity <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a\rightarrow (b\rightarrow c) = b\rightarrow (a\rightarrow c)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">→</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">→</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We give a description of the implicative-infimum subreducts of the commutative weak Heyting algebras. Moreover, we present an equivalence for the algebraic category of commutative weak Heyting algebras, which is based in a Priestley-style duality.</p>

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On commutative weak Heyting algebras and some of its subreducts

  • 期刊论文

  • 发表日期：

    2026年3月25日

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  • Sergio A. Celani,
  • Hernán J. San Martín

摘要

The variety of weak Heyting algebras corresponds to the strict implication fragment of the normal modal logic \(\textrm{K}\) K , which is also known as the subintuitionistic local consequence of the class of all Kripke models.

In this paper we study the variety of commutative weak Heyting algebras, which is defined as the subvariety of weak Heyting algebras whose members satisfy the identity \(a\rightarrow (b\rightarrow c) = b\rightarrow (a\rightarrow c)\) a → ( b → c ) = b → ( a → c ) . We give a description of the implicative-infimum subreducts of the commutative weak Heyting algebras. Moreover, we present an equivalence for the algebraic category of commutative weak Heyting algebras, which is based in a Priestley-style duality.