Fuzzy relational compositions; Compositions of partial fuzzy relations; Partial fuzzy set theory; Undefined values; Missing values; Indeterminable values

Four-valued logics were developed from three-valued logics aiming at creating tools dealing with missing values and indeterminable values at the same time in computer science. In this paper, we are motivated by the formation of such four-valued logics and construct a new set of operations able to work with both types of such undefined values in partial fuzzy set theory. The operations are built based on the well-known Bochvar algebra and the Sobocinski algebra modeling indeterminable values, and the recent Dragonfly algebra purely designed for modeling missing values. Moreover, the operations are established to be compatible with the operations used for elaborating compositions of partial fuzzy relations. Various valid properties of the new operations are then presented, and consequently, the properties of the compositions of partial fuzzy relations constructed based on such operations are implied as well. In the end, an illustrative example is provided to observe the use of the proposed compositions.