<p>In this paper, we investigate two extensions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals in state monadic MV-algebras, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> is a complete Heyting algebra. First, we introduce <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy congruences in state monadic MV-algebras, show that there exists one-to-one correspondence between the set of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals and the set of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy congruences. Then, we study the type-I extension of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals and obtain the set of all type-I extension of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals is a complete Heyting algebra. In addition, we give the definition of the type-II extension of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals, and prove that the set of all stable <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals relative to an <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy set is a complete Heyting algebra and the set of all involutory <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals relative to an <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideal is a complete Boolean algebra, respectively. Most importantly, we use these two extensions of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals to give a description of any <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideal via different construction methods. Finally, we consider the relationship between the type-I extension and the type-II extension of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-fuzzy state monadic ideals.</p>

Two extensions of $\mathcal {L}$-fuzzy state monadic ideals in state monadic MV-algebras

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发表日期：

2026年2月3日

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Ya Wei

摘要

In this paper, we investigate two extensions of $\mathcal {L}$ $L$ -fuzzy state monadic ideals in state monadic MV-algebras, where $\mathcal {L}$ $L$ is a complete Heyting algebra. First, we introduce $\mathcal {L}$ $L$ -fuzzy state monadic ideals and $\mathcal {L}$ $L$ -fuzzy congruences in state monadic MV-algebras, show that there exists one-to-one correspondence between the set of $\mathcal {L}$ $L$ -fuzzy state monadic ideals and the set of $\mathcal {L}$ $L$ -fuzzy congruences. Then, we study the type-I extension of $\mathcal {L}$ $L$ -fuzzy state monadic ideals and obtain the set of all type-I extension of $\mathcal {L}$ $L$ -fuzzy state monadic ideals is a complete Heyting algebra. In addition, we give the definition of the type-II extension of $\mathcal {L}$ $L$ -fuzzy state monadic ideals, and prove that the set of all stable $\mathcal {L}$ $L$ -fuzzy state monadic ideals relative to an $\mathcal {L}$ $L$ -fuzzy set is a complete Heyting algebra and the set of all involutory $\mathcal {L}$ $L$ -fuzzy state monadic ideals relative to an $\mathcal {L}$ $L$ -fuzzy state monadic ideal is a complete Boolean algebra, respectively. Most importantly, we use these two extensions of $\mathcal {L}$ $L$ -fuzzy state monadic ideals to give a description of any $\mathcal {L}$ $L$ -fuzzy state monadic ideal via different construction methods. Finally, we consider the relationship between the type-I extension and the type-II extension of $\mathcal {L}$ $L$ -fuzzy state monadic ideals.

Two extensions of \(\mathcal {L}\)-fuzzy state monadic ideals in state monadic MV-algebras

摘要