−Table of Contents
Idempotent semirings with identity and zero
Abbreviation: ISRng0101
Definition
An \emph{idempotent semiring with identity and zero} is a semirings with identity and zero S=⟨S,∨,0,⋅,1⟩S=⟨S,∨,0,⋅,1⟩ such that ∨∨ is idempotent: x∨x=xx∨x=x
Morphisms
Let SS and TT be idempotent semirings with identity and zero. A morphism from SS to TT is a function h:S→Th:S→T that is a homomorphism:
h(x∨y)=h(x)∨h(y)h(x∨y)=h(x)∨h(y), h(x⋅y)=h(x)⋅h(y)h(x⋅y)=h(x)⋅h(y), h(0)=0h(0)=0, h(1)=1h(1)=1
Examples
Example 1:
Basic results
Properties
Finite members
f(1)=1f(2)=1f(3)=3f(4)=20f(5)=149f(6)=1488f(7)=18554