Fuzzy Logic Systems for the Weakest Uninorm

The weakest uninorm $\underline{\mathcal{U}}_{g}$ is conjunctive but not left-continuous, and there is not residual implicator. We studied the weakest uninorm logic UDP, which is an extension of UL. It proved that logic UDP is canonically complete. The finite embeddability property of the corresponding semantic model is investigated. In addition, according to the ordinal sum theorem of semigroup and ordinal sum construction of uninorm, a new left-continuous and conjunctive uninorm is constructed, which is a revision of the smallest uninorm. Moreover, we shown that the logic is standard completeness.