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.

SUBMITTER: Wu J 

PROVIDER: S-EPMC4147286 | biostudies-other | 2014

REPOSITORIES: biostudies-other