{"database":"biostudies-other","file_versions":[],"scores":null,"additional":{"omics_type":["Unknown"],"volume":["2014"],"submitter":["Wu J"],"journal":["The Scientific World Journal"],"pagination":["917432"],"full_dataset_link":["https://www.ebi.ac.uk/biostudies/studies/S-EPMC4147286"],"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."],"repository":["biostudies-other"],"data_source":["Europe PMC"],"pubmed_authors":["Wu J"],"additional_accession":[]},"is_claimable":false,"name":"Groups of negations on the unit square.","description":"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.","dates":{"release":"2014-01-01T00:00:00Z","publication":"2014 ","modification":"2019-08-04T07:58:25Z","creation":"2019-08-04T07:58:25Z"},"accession":"S-EPMC4147286","cross_references":{"DOI":["10.1155/2014/917432 "]}}