THE WEAK SUBSTITUTION IN COMPLEX NUMBERS OVER THE SYMMETRIZED MAX-PLUS ALGEBRA

Abstract

The complex numbers over the symmetrized max-plus algebra is an extension of numbers in the symmetrized max-plus algebra. This complex number is formed by defining an imaginary form in the symmetrized max-plus algebra. The substitution process in complex numbers over the symmetrized max-plus algebra is a very important concept and cannot be done as in complex numbers over the conventional algebra. Thus, several algebraic properties are needed to ensure that the substitution process can be carried out. In this paper, we show the substitution rules in complex numbers over the symmetrized max-plus algebra. These rules is formulated by extending the weak substitution property in the symmetrized max-plus algebra. The extension was carried out in the symmetrized max-plus algebra part and imaginary part of complex numbers. The results obtained are the weak substitution in complex numbers over the symmetrized max-plus algebra. Substitution process can be carried out in complex numbers if both of the symmetrized max-plus algebra part and imaginary part of complex numbers are signed elements. Then, according the weak substitution property, it can be derived the weak transitive property in complex numbers over the symmetrized max-plus algebra. Furthermore, it can be also derived reduction of balance property in complex numbers over the symmetrized max-plus algebra, which reduce balance relation into equal relation sense.