On Non-commutative Residuated Lattices With Internal States

Abstract

In this paper, we enlarge the language of non-commutative residuated lattices to provide a unified algebraic foundation for probabilities of fuzzy events in substructural logics, by adding an internal state that describes algebraic properties of states. The resulting class of algebras will be called non-commutative residuated lattices with internal states (or state residuated lattices for short). First, we prove that any perfect residuated lattice admits a nontrivial internal state and discuss some algebraic properties of internal states. Also, we give characterizations of divisible residuated lattices and idempotent residuated lattices, and obtain relationships between internal states and states on residuated lattices. Moreover, using some kinds of state filters, we present some characterizations of local state residuated lattices and their subclasses. Furthermore, we obtain that each local state commutative residuated lattice is either perfect or locally finite or peculiar. Finally, we prove that the class $SF[L]$ of all state filters in state residuated lattices is a complete Heyting algebra. In particular, by studying the state co-annihilator of a nonempty set with respect to a state filter, we prove that 1) the class $S_{X}SF[L]$ of all stable state filters relative a nonempty set $X$ in state residuated lattices is also a complete Heyting algebra, but it is not a subalgebra of the Heyting algebra $SF[L]$ ; 2) the class $I_{F}SF[L]$ of all involutory state filters relative a state filter $F$ in state residuated lattices is a complete Boolean algebra.