A survey of fuzzy implication algebras and their axiomatization

• Fuzzy implication algebras are important in fuzzy logic.
• Studies on FI-algebras are reviewed.
• Axiomatization of FI-algebras and their important subclasses are shown.
• The relationships between FI-algebras and other fuzzy algebras are discussed.

The theory of fuzzy implication algebras was proposed by Professor Wangming Wu in 1990. The present paper reviews the following two aspects of studies on FI-algebras: concepts, properties and some subclasses of FI-algebras; axiomatization of the class of FI-algebras and some of its important subclasses. The main results are summarized in the current paper, the relationships between FI-algebras and several classes of important fuzzy algebras are discussed, such as BL-algebras, MTL-algebras, and residuated lattices, and propositional calculus systems of several special classes of FI-algebras are shown.