Groups of Negations on the Unit Square

Abstract EN

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 LI, and the other group contains all the isomorphisms and all the strict negations on LI, 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.