On filters of non-associative residuated lattices (commutative residuated lattice-ordered groupoids)

Filter theory of non-associative residuated lattices (i.e., commutative residuated lattice-ordered groupoids) are constructed. Some properties of filters of non-associative residuated lattices are given, and the quotient algebraic structure by filter is established. Moreover, the notion of Boolean filter is introduced, and some necessary and sufficient conditions are proved. Finally, a necessary and sufficient condition for a commutative residuated lattice-ordered groupoid to be a residuated lattice is presented, and a characterization of Boole algebra in residuated lattice is obtained by the notion of Boolean filter.