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<p>In the literature many generalizations of Boolean algebras exist viz. Ockham algebras, De Morgan algebras, p-algebras, Heyting algebras etc. There has been investigations into algebras in which two or more of such negations occur simultaneously. This paper investigates the class of algebras called quad algebras which encompasses both the Boolean and De Morgan algebras. Due to the presence of Boolean negation such algebras naturally possess a ring structure. In fact these algebras turn out to be equivalent with the class of rings where every element satisfies the polynomial equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10204_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(x^{4}=x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>. On the other aspect, we provide a 4-valued semantics of the logic for quad algebras. The propositional logic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11225_2025_10204_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {L}_{\mathcal{Q}\mathcal{A}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mrow> <mi mathvariant="script">Q</mi> <mi mathvariant="script">A</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of a quad algebras is shown to be sound and complete with respect to a 4-valued semantics.</p>

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Quad Rings, Quad Algebras and a 4-valued Logic

  • 期刊论文

  • 发表日期：

    2025年9月5日

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  • Arun Kumar,
  • Bisham Dewan,
  • Neha Gaur

摘要

In the literature many generalizations of Boolean algebras exist viz. Ockham algebras, De Morgan algebras, p-algebras, Heyting algebras etc. There has been investigations into algebras in which two or more of such negations occur simultaneously. This paper investigates the class of algebras called quad algebras which encompasses both the Boolean and De Morgan algebras. Due to the presence of Boolean negation such algebras naturally possess a ring structure. In fact these algebras turn out to be equivalent with the class of rings where every element satisfies the polynomial equation \(x^{4}=x\) x 4 = x . On the other aspect, we provide a 4-valued semantics of the logic for quad algebras. The propositional logic \(\mathcal {L}_{\mathcal{Q}\mathcal{A}}\) L Q A of a quad algebras is shown to be sound and complete with respect to a 4-valued semantics.