Parametric families of fuzzy consequence operators

In a previous paper ([6]) we explored the notion of coherent fuzzy consequence
operator. Since we did not know of any example in the literature of
non-coherent fuzzy consequence operator, we also showed several families of
such operators. It is well-known that the operator induced by a fuzzy preorder
through Zadeh’s compositional rule is always a coherent fuzzy consequence
operator. It is also known that the relation induced by a fuzzy consequence
operator is a fuzzy preorder if such operator is coherent ([5]). The aim of
this paper is to show a parametric family of non-coherent fuzzy consequence
operators which induce a preorder and also a family of non-coherent fuzzy
consequence operators which do not induce a preorder. These families of
operators can be implemented through very simple algorithms.