A non-associative generalization of continuous t-norm-based logics

This paper investigates a non-associative generalization, more exactly, a weak t-associative (wta) generalization, of continuous t-norm-based logics. First, the wta-uninorm logic WA_tBL and its axiomatic extensions WA_t, WA_tΠ, and WA_tG are introduced as [0, e]-continuous wta-uninorm-based analogues of the most famous continuous t-norm-based logics, BL (Basic fuzzy logic), (ukasiewicz logic), Π (Product logic), and G (Gödel logic), respectively. Their corresponding algebraic structures and algebraic completeness results are considered. Next, completeness with respect to the algebras whose lattice reduct is [0, 1], known as standard completeness, is established for these systems by a construction in the style of Jenei-Montagna.