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<p>A <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\nabla \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [<CitationRef CitationID="CR1">1</CitationRef>], we explored the algebraic properties of various varieties of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\nabla \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\nabla \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\nabla \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\nabla \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∇</mi> </math></EquationSource> </InlineEquation>-algebras.</p>

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On a Generalization of Heyting Algebras II

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年7月31日

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  • Amirhossein Akbar Tabatabai,
  • Majid Alizadeh,
  • Masoud Memarzadeh

摘要

A \(\nabla \) ∇ -algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [1], we explored the algebraic properties of various varieties of \(\nabla \) ∇ -algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce \(\nabla \) ∇ -spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of \(\nabla \) ∇ -algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of \(\nabla \) ∇ -algebras.