Subdistributivity and superdistributivity of binary operations

Recently, the property of distributivity of binary operations on [0,1] is of great interest both for practical as well as fundamental reasons. In this paper we summarize some facts connected with distributivity and non-distributivity of binary operations on [0,1]. Motivated by non-distributive logical connectives in fuzzy set theory we show some new results concerning the above property. A new direction of investigations mainly concerns the distributivity inequalities for triangular norms and conorms and their diverse generalizations. In particular, we look for examples of binary operations fulfilling the distributivity inequalities.