A complete invariant system for noetherian BL-algebras and more general L-algebras

Abstract

Main results on BL-algebras, including their classification in the finite case, are reconsidered and extended to a class of L-algebras X with prime factorization, including BL-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite BL-algebra reappears as a canonical L-subalgebra $\tilde{P}\left(X\right)$ of prime elements in the self-similar closure $S\left(X\right)$ where $\tilde{P}\left(X\right)$ is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in X. These invariants determine X within its self-similar closure $S\left(X\right)=S\left(\tilde{P}\right(X\left)\right)$. The three basic types of BL-algebras are related to concepts of L-algebras with further-reaching significance in quantum theory.