2-Dimension Linguistic Lattice Implication Algebra

This paper concerns the representation of the linguistic evaluation set, which is often a total order set for its simple structure and operations. However, in some uncertain situations, the linguistic evaluation set with total order set is not enough for providing the assessments of the decision maker because all the linguistic labels are comparable. For describing the comparable and incomparable linguistic linguistic evaluation labels, a 2-dimension linguistic lattice implication algebra (2DL-LIA) is firstly introduced by combining two linguistic evaluation sets and a lattice implication algebra (LIA). Subsequently, the comparison analysis between two labels of a 2DL-LIA is investigated according to the structure of lattice. Finally, some operators are defined by using of the logical operators in a 2DL-LIA, and some properties of these operators are discussed, which are quite useful in the aggregation process of multi-attribute decision making (MADM).