Notice of RetractionKronecker products of lattice-valued finite automata

Notice of Retraction

After careful and considered review of the content of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE´s Publication Principles.

We hereby retract the content of this paper. Reasonable effort should be made to remove all past references to this paper.

The presenting author of this paper has the option to appeal this decision by contacting TPII@ieee.org.

In order to solve the structure problem of product automata, the matrix theory is used in this paper. By introducing Kronecker product, product structure of automata can be translated into matrix product. Applying lattice-fuzzy matrix theory, the concepts of lattice-fuzzy transition matrixes, lattice-valued transformation matrix semigroups, as well as coverings for lattice-valued finite automata are introduced. The equivalence relation is defined in the set of input symbols. For each lattice-valued finite state automaton, we have showed that there exists a lattice-valued transformation matrix semigroup associated with it. The definitions of products of lattice-valued fuzzy finite state machines are given by application of Kronecker product. Furthermore, the properties of lattice-fuzzy transition matrix for three kinds Kronecker products of lattice-valued fuzzy finite state machines are discussed. The covering relationships and associative properties among Kronecker products of lattice-valued transformation matrix semigroup associated with lattice-valued fuzzy finite state machines are studied. These results show that Kronecker product is compatible with the product of lattice-valued finite automata. Also Kronecker product can effectively describe and simplify the product structure of automata.