Groups of Negations on the Unit Square

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 G (2) (or G (3) or G (4), correspondingly). All these four kinds of mappings form a group G (5). And all the groups G (i), i = 2,3, 4 are normal subgroups of G (5). Moreover, for G (5), 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.