Prime spectrums of EQ-algebras.

This article focuses on the study of prime spectrum spaces of EQ-algebras, particularly good EQ-algebras, and addresses two open problems concerning |$\wedge$|-prime spectrums of involutive and prelinear EQ-algebras posed by N. Akhlaghinia, R. A. Borzooei, and M. A. Kologani. The authors provide characterizations of preideals, prime preideals, and maximal preideals, introduce quasi De Morgan EQ-algebras (MEQ-algebras), and prove that for MEQ-algebras, prime preideals coincide with |$\wedge$|-prime preideals. They establish that the prime spectrum space of a good EQ-algebra is compact and that for involutive EQ-algebras, this space is connected if and only if the Boolean center has exactly two elements. Additionally, they show that the maximal spectrum space of good and prelinear EQ-algebras (including involutive and prelinear ones) is a normal Hausdorff space regardless of cardinality, thus resolving the two open problems. Finally, the paper characterizes the spectrum space of MEQ-algebras via their reticulation, linking algebraic and topological structures.