Prelinear and Involution Non-associative Residuated Lattices
Xiaohong Zhang · Mohu xitong yu shuxue · 2012
Non-associative residuated lattice is a common algebraic abstract of various non-associative lattice-valued logic systems.In this paper,the algebraic properties of several special non-associative residuated lattices are studied.It is proved that every prelinear non-associative residuated lattice is a bounded distributive lattice,and a necessary and sufficient condition for non-associative residuated lattice to be prelinear is given.The notions of involution and strong involution non-associative residuated lattice are introduced,and their basic properties are studied.Moreover,the equivalence conditions for involution non-associative residuated lattices and strong involution non-associative residuated lattices are given respectively.Finally a counter example is given to show that there is a strong involution and prelinear non-associative residuated lattice which is not an implication lattice.