What would be the simplest way to define the semiring with elements $\{0, 1\}$ with the logical OR ($|$) operation as addition and the standard integer multiplication?
At very least it should be accepted by PolynomialRing as the domain for coefficients.
OPTION #1. Luckily this semiring is a subsemiring of InfinityRing:
sage: from sage.matrix.operation_table import OperationTable
sage: R = InfinityRing
sage: OperationTable([R(0),R(1)], operation=operator.add, names='digits')
+ 0 1
+----
0| 0 1
1| 1 1
sage: OperationTable([R(0),R(1)], operation=operator.mul, names='digits')
* 0 1
+----
0| 0 0
1| 0 1
There is however a bug in addition, which I reported at https://trac.sagemath.org/ticket/34231
OPTION #2. Another approach is to use TropicalSemiring as follows:
R = TropicalSemiring(GF(2), use_min=False)
Asked: 2022-07-11 22:45:31 +0100
Seen: 198 times
Last updated: Oct 17 '22
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Is this
GF(2)?No. In $GF(2)$ we have $1+1=0$, while here $1+1=1$.