Basic Semilinear Logics Based on [0, u]-continuous Uwa-uninorms

This paper introduces micanorm-based logics satisfying three forms of weak associativity as a weak associative generalization of the [0, 1)-continuous uninorm-based logic BUL introduced by Gabbay and Metcalfe. To be more precise, we first introduce the basic uwa-uninorm logic WA_U BUL and its axiomatic extensions A_U BUL, SA_U BUL as u-weak-associative uninorm analogues of the basic uninorm logic BUL. We then deal with algebraic completeness results for them by introducing their corresponding algebraic structures. Next, we introduce uwa-uninorms as uninorms satisfying weak u-associativity in place of associativity and study related algebraic properties. We finally show that the uwa-uninorm logics are standard complete, that is, complete on unit interval [0, 1], using Yang–style construction.