Research On Distributivity About Quasi-arithmetic Means And 2-uninorms

At first,this paper characterized the distributivity equations between quasi-arithmetic means and some binary operators,such as,nullnorms,semi-nullnorms and semi-t-operators.On one hand,it is shown that the distributivity for nullnorms(semi-nullnorms)over quasi-arithmetic means degenerate into the distributivity for t-norms or t-conorms(semi-t-norms or semi-t-conorms)over quasi-arithmetic means.But about the distributivity for semi-t-operators over quasi-arithmetic means,we have new solutions G(x,y)= x or G(x,y)= y.On the other hand,when quasi-arithmetic means are left or right distributive over semi-nullnorms(nullnorm-s),which not only degenerate into semi-t-norms or semi-t-conorms(t-norms or t-conorms),but also are idempotent.About the left or right distributivity for quasi-arithmetic means over semi-t-operators,which have new solutions G(x,y)= x or G(x,y)= y except degenerate into idempotent t-norms or idempotent t-conorms.These new results extend the previous ones about the distributivity between t-norms(t-conorms)and quasi-arithmetic means.What’s more,this paper absolutely characterized the distributivity equations for 2-uninorms over semi-uninorms.2-uninorm covers both a uninorm and a nullnorm,which forms a class of commutative,associative and increasing operators on the unit interval with an absorbing el-ement separating two subintervals having their own neutral elements.Based on the values of G(0,1)、G(1,k)and G(0,k),2-uninorms are divided into five subclasses.Through discussing the distributivity of each kind of 2-uninorms over semi-uninorms respectively,we not only determine the specific structure of these semi-uninorms,but also characterize the structure of each kind of 2-uninorms.These results in this work are complementary to the distributivity equation in the class of 2-uninorms.