On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra

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Background In this paper, we explore the Cartesian product of fuzzy PUP ideals and subalgebras within a PUP (pseudo-UP)algebra X. We investigate how fuzzy pseudo-UP ideals are affected under PUP homomorphisms, particularly focusing on the image and inverse image of the fuzzy sets K α and K v , which generalize classical fuzzy sets. These fuzzy sets aim to quantify the level of ambiguity in fuzzy situations. Methods We introduced the concept of the Cartesian product of fuzzy PUP ideals and subalgebras of PUP algebra and analyzed their properties. Specifically, we studied the behavior of fuzzy pseudo-UP ideals under PUP homomorphism, using the fuzzy sets K α and K v to model ambiguity. We also examined the strongest fuzzy PUP relation on a fuzzy PUP subalgebra and ideal of PUP algebra. Results We proved that the Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal. This result was characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure. Additionally, the strongest fuzzy PUP relation on a fuzzy PUP subalgebra was defined, and several important properties were derived. Conclusions This work extends the concept of fuzzy PUP ideals in the context of PUP algebras and explores how fuzzy homomorphism influence their structure. Our results contribute to a better understanding of fuzzy relations in algebraic settings and open the door for further research into the properties and applications of fuzzy ideals in PUP algebras.

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