On state ideals and state relative annihilators in De Morgan state residuated lattices
Summary: The concept of state has been considered in commutative and noncommutative logical systems, and their properties are at the center for the development of an algebraic investigation of probabilistic models for those algebras. This article mainly focuses on the study of the lattice of state ideals in De Morgan state residuated lattices (DMSRLs). First, we prove that the lattice of all state ideals \(\mathcal{SI}\left( L\right)\) of a DMSRL \(\left( L, \tau\right)\) is a coherent frame. Then, we characterize the DMSRL for which the lattice \(\mathcal{SI}\left( L\right)\) is a Boolean algebra. In addition, we bring in the concept of state relative annihilator of a given nonempty subset with respect to a state ideal in DMSRL and investigate various properties. We prove that state relative annihilators are a particular kind of state ideals. Finally, we investigate the notion of prime state ideal in DMSRL and establish the prime state ideal theorem.
scientific article; zbMATH DE number 7595553
scientific article