Generalization of extensive structures and its representation

This paper generalizes extensive structures so that a weighted additive model can be obtained. A left nonnegative concatenation structure with left identity is defined as a nonnegative concatenation structure (Luce et al., 1990) with left identity for which the solvability and Archimedean properties are satisfied only related to left-concatenation. This structure has two partial binary operations-multiplication and right division-and a new partial binary operation is defined on it. Two conditions of equivalence form are then provided to make the left nonnegative concatenation structure with left identity into an extensive structure with identity with respect to the newly defined operation. Finally, the weighted additive model is derived from an additive representation on the extensive structure, so that distinct m-period and n-period (m≠n) temporal sequences can be compared.