A max-plus algebra is a semiring over the union of real numbers and
ε
=
−
∞
, equipped 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.
Let a and b be real scalars or ε. Then the operations maximum (implied by the max operator
⊕
) and addition (plus operator
⊗
) for these scalars are defined as
a
⊕
b
=
max
(
a
,
b
)
a
⊗
b
=
a
+
b
Watch: Max-operator
⊕
can easily be confused with the addition operation. Similar to the conventional algebra, all
⊗
- operations have a higher precedence than
⊕
- operations.
Max-plus algebra can be used for matrix operands A, B likewise, where the size of both matrices is the same. To perform the A
⊕
B - operation, the elements of the resulting matrix at (row i, column j) have to be set up by the maximum operation of both corresponding elements of the matrices A and B:
[
A
⊕
B
]
i
j
=
[
A
]
i
j
⊕
[
B
]
i
j
=
max
(
[
A
]
i
j
,
[
B
]
i
j
)
The
⊗
- operation is similar to the algorithm of Matrix multiplication, however, every "+" calculation has to be substituted by an
⊕
- operation and every "
⋅
" calculation by a
⊗
- operation. More precisely, to perform the A
⊗
B - operation, where A is a m×p matrix and B is a p×n matrix, the elements of the resulting matrix at (row i, column j) are determined by matrices A (row i) and B (column j):
[
A
⊗
B
]
i
j
=
⨁
k
=
1
p
(
[
A
]
i
k
⊗
[
B
]
k
j
)
=
max
(
[
A
]
i
1
+
[
B
]
1
j
,
…
,
[
A
]
i
p
+
[
B
]
p
j
)
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:
ε
⊕
a
=
a
ε
⊗
a
=
ε
To point the zero number out, the element e was defined by
e
=
0
. Therefore:
e
⊗
a
=
a
.
Obviously, ε is the neutral element for the
⊕
-operation, as e is for the
⊗
-operation
associativity:
commutativity :
distributivity:
zero element :
unit element:
idempotency of
⊕
:
Butkovič, Peter (2010), Max-linear Systems: Theory and Algorithms, Springer Monographs in Mathematics, Springer-Verlag, doi:10.1007/978-1-84996-299-5
