On ⊙-Ideals and Lattices of ⊙-Ideals in Regular Residuated Lattices

In this paper, the operation circle dot and the concept of circle dot-ideals of (regular) residuated lattices are introduced. Some characterization theorems for circle dot-ideals of (regular) residuated lattices are given. Representation theorems about circle dot-ideals which are generated by non-empty subsets of regular residuated lattices are obtained. For the set of all circle dot-ideals of a (regular) residuated lattice, an adjunction pair is defined. It is proved that the lattice of all circle dot-ideals in a regular residuated lattice with the adjunction and the set-inclusion order is a complete Heyting algebra (i.e., a frame) and an algebraic lattice, which thus gives a new distributive residuated lattice.