title: Groups of negations on the unit square.
abstract: The main results are about the groups of the negations on the unit square, which is considered as a bilattice. It is proven that all the automorphisms on it form a group; the set, containing the monotonic isomorphisms and the strict negations of the first (or the second or the third) kind, with the operator "composition," is a group G₂ (or G₃ or G₄, correspondingly). All these four kinds of mappings form a group G₅. And all the groups Gi , i = 2,3, 4 are normal subgroups of G₅. Moreover, for G₅, a generator set is given, which consists of all the involutive negations of the second kind and the standard negation of the first kind. As a subset of the unit square, the interval-valued set is also studied. Two groups are found: one group consists of all the isomorphisms on L(I) , and the other group contains all the isomorphisms and all the strict negations on L(I) , which keep the diagonal. Moreover, the former is a normal subgroup of the latter. And all the involutive negations on the interval-valued set form a generator set of the latter group.
authors.affiliation: Department of Mathematics, Shandong Normal University, Jinan 250014, China.
authors.forename: Jiachao
authors.identifier: 0000-0002-5734-0837
authors.initials: J
authors.lastname: Wu
mesh_terms: D001185:Artificial Intelligence; D008128:Logic; D055641:Mathematical Concepts; D008962:Models, Theoretical