A max-plus algebra is an algebra over the real numbers with maximum and addition as the two binary operations. It can be used appropriately to determine marking times within a given Petri net and a vector filled with marking state at the beginning.
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Let A and B be scalars. Then the operations maximum (implied by the max operator ) and addition (plus operator ) for this scalars are defined as
Watch: Max-operator can easily be confused with the addition operation. All - operations have a higher precedence than - operations.
Max-Plus algebra can be used for matrix operands M, N likewise. To perform the - operation, the elements of the resulting matrix R (row i, column j) have to be set up by the maximum operation of both corresponding elements of the matrices M and N:
The - operation is similar to algorithm of Matrix multiplication, however, every "+" calculation has to be substituted by a - operation, every "" calculation by a - operation.
In order to handle marking times like which means "never before", the ε-element has been established by ε. According to the idea of infinity, the following equations can be found:
To point the zero number out, the element e was defined by . Therefore:
Obviously, ε is the neutral element for the - operation as well as e is for the - operation
Note: In general, A B = B A does not hold, for example in the case of matrix operations.