The Convolution Operators On Fuzzy Truth Values

This paper mainly studies the lattice structure for the algebra of fuzzy truth values and the basic properties of convolution operators on fuzzy truth values,such as idempotency,distributivity between extended operators,distributivity of convolution operators over meet-convolution and join-convolution,the boundary conditions,etc.And on this basis,uninorms and nullnorms on fuzzy truth values are constructed.Firstly,we give a necessary and sufficient condition for the subalgebras of the algebra of fuzzy truth values to form lattices,and abbreviate the subalgebras that form lattices as sublattices.Furthermore,it is proved that the mapping that makes any fuzzy truth value convex is a monomorphism from any given sublattice to an appropriate complete sublattice,where the complete sublattice is either composed of all normal convex fuzzy truth values or differs only in height.Secondly,we describe the convolution operator on fuzzy truth values that satisfies the idempotence.We give the sufficient conditions and necessary conditions for various fuzzy truth values to satisfy the idempotence equation.In particular,the necessary and sufficient condition for set fuzzy truth values,convex fuzzy truth values and general fuzzy truth values to satisfy the idempotence equation is given,respectively.Thirdly,we study the distributivity between extended operators.We give two sufficient conditions under which the distributive equation holds at any interval segment,and then obtain a sufficient condition of which the distributive equation holds and a necessary and sufficient condition of which the distributive equation holds under certain preconditions.At the same time,the distributivity of two types of extended uninorms over extended operators is studied,and then a sufficient condition under which these two types of extended uninorms distribute over extended nullnorms is obtained.Fourthly,we study the distributivity of convolution operators over meetconvolution and join-convolution.On the one hand,we focus on whether various fuzzy truth values satisfy the two distributive equations,and then we obtain the sufficient condition under which singleton and interval fuzzy truth values satisfy the distributive equations,respectively.On the other hand,we successively focus on the distributivity of max-convolution,min-convolution and extended operators over meet-convolution and join-convolution,and then we obtain some sufficient conditions under which the corresponding distributive equations hold and a necessary and sufficient condition of which extended operators distribute over meet-convolution and join-convolution under certain preconditions.In particular,we give a necessary and sufficient condition under which join-convolution is distributive over meet-convolution,and similarly give a necessary and sufficient condition under which meet-convolution is distributive over join-convolution.Finally,we give the sufficient conditions for convolution operators to satisfy the commutativity,associativity,monotonicity and some boundary properties,as well as the necessary and sufficient conditions under certain preconditions.Based on the above work,we use extended operators and general extended operators to construct three kinds of uninorms and nullnorms on several types of fuzzy truth values.