Uni-nullnorms and null-uninorms on bounded lattices and their properties

Uninorms and nullnorms are generalizations of t-norms and t-conorms that have been extensively studied and are extremely useful for many applied sciences such as fuzzy quantifiers, expert systems, and neural networks([4,7]). Uninorms in the unit real interval were defined by Yager and Rybalov ([3]) in 1996 and many studies were conducted on uninorms in the unit real interval. In 2015, uninorms on bounded lattices were defined and their properties were investigated by Karaçal and Mesiar([8]). Nullnorms were first studied by Mas, Mayor and Torrens([4]) in 1999 and by Calvo, De Baets and Fodor ([6]) in 2001. In these studies, the properties of nullnorms on the unit real interval are given. The first study on bounded lattices was done by Karaçal, İnce and Mesiar ([9]) in 2015. In this study, the largest and smallest nullnorms with a zero element are obtained on a bounded lattice L. After this study, many researchers examined nullnorms defined on bounded lattices([8,11,12]). Uni-nullnorms and null-uninorms; They are very important functions, which are the generalization of nullnorms and uninorms, defined by Sun, Wang, and Qu ([10]) in 2017. In this study, it is shown that uni-nullnorms and null-uninorms are dual pairs of each other on the unit real interval. Then, uni-nullnorms on bounded lattices were studied by Wang, Zhan and Liu ([11]) in 2019, and in their study, the definition and properties of uni-nullnorms on an L bounded lattice are given. In this study, some properties of uni-nullnorms and null-uninorms will be examined on bounded lattices. In addition, it is aimed to determine the smallest and largest of {0,e}_a 2-neutral element null-uninorms and {e,1}_a 2-neutral element uni-nullnorms on an arbitrary bounded lattice L. For this, firstly the basic properties of lattices, t-norms (t-conorms), uninorms and nullnorms will be mentioned.