Recently, Bandaru et al. and Borumand Saeid et al. introduced transitive GE-algebras (for short, TGE-algebras) and antisymmetric transitive GE-algebras (for short, ATGE-algebras), respectively. In this paper, we present TGE-logic, the corresponding logic to these algebras. We show that the variety of TGE-algebras forms an algebraic semantics for TGE-logic in the sense of (Blok & Pigozzi, <CitationRef CitationID="CR2">1989</CitationRef>). Furthermore, we prove that TGE-logic is strongly sound and complete with respect to TGE-algebras. We demonstrate that the equivalent algebraic semantics for TGE-logic is the variety of ATGE-algebras and that TGE-logic is algebraizable in the sense of (Blok & Pigozzi, <CitationRef CitationID="CR2">1989</CitationRef>). Moreover, we prove that TGE-logic is strongly sound and complete with respect to ATGE-algebras. We further prove each congruence relation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> on an ATGE-algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {L}}= (L, *, 1) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mrow /> <mo>∗</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is uniquely determined by the equivalence class of 1 under <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>, where the quotient algebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{{\mathcal {L}}}{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi mathvariant="script">L</mi> <mi>θ</mi> </mfrac> </math></EquationSource> </InlineEquation> is an ATGE-algebra. Finally, we show the independence of the axioms of TGE-logic.

Logic for transitive GE-algebras

期刊论文
发表日期：

2026年6月25日

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Masoud Haveshki,
Mahboobeh Mohamadhasani

摘要

Recently, Bandaru et al. and Borumand Saeid et al. introduced transitive GE-algebras (for short, TGE-algebras) and antisymmetric transitive GE-algebras (for short, ATGE-algebras), respectively. In this paper, we present TGE-logic, the corresponding logic to these algebras. We show that the variety of TGE-algebras forms an algebraic semantics for TGE-logic in the sense of (Blok & Pigozzi, 1989). Furthermore, we prove that TGE-logic is strongly sound and complete with respect to TGE-algebras. We demonstrate that the equivalent algebraic semantics for TGE-logic is the variety of ATGE-algebras and that TGE-logic is algebraizable in the sense of (Blok & Pigozzi, 1989). Moreover, we prove that TGE-logic is strongly sound and complete with respect to ATGE-algebras. We further prove each congruence relation $\theta $ $θ$ on an ATGE-algebra ${\mathcal {L}}= (L, *, 1) $ $L = (L, *, 1)$ is uniquely determined by the equivalence class of 1 under $\theta $ $θ$ , where the quotient algebra $\frac{{\mathcal {L}}}{\theta }$ $\frac{L}{θ}$ is an ATGE-algebra. Finally, we show the independence of the axioms of TGE-logic.