A
maxplus algebra is an
algebraIn mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
over the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s with maximum and addition as the two binary operations.
It can be used appropriately to determine marking times within a given
Petri netA Petri net is one of several mathematical modeling languages for the description of distributed systems. A Petri net is a directed bipartite graph, in which the nodes represent transitions and places...
and a vector filled with marking state at the beginning.
Scalar operations
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: Maxoperator
can easily be confused with the addition operation. All
 operations have a higher
precedenceIn mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....
than
 operations.
Matrix operations
MaxPlus 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:
 R_{ij} = M_{ij} N_{ij}
The
 operation is similar to algorithm of
Matrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an nbym matrix and B is an mbyp matrix, the result AB of their multiplication is an nbyp matrix defined only if the number of columns m of the left matrix A is the...
, however, every "+" calculation has to be substituted by a
 operation, every "
" calculation by a
 operation.
Useful enhancement elements
In order to handle marking times like
which means "never before", the εelement has been established by ε
. According to the idea of infinity, the following equations can be found:
 ε A = A
 ε A = ε
To point the zero number out, the element
e was defined by
. Therefore:
 e A = A
Obviously, ε is the neutral element for the
 operation as well as
e is for the
 operation
Algebra properties








Note: In general, A
B = B
A does not hold, for example in the case of matrix operations.