Multiple definitions of Semirings

Question

I am currently studying for my algebra exam and came across the definition of a semiring. Reading through multiple books at once to better understand the definitions and examples I encountered different definitions of semirings.

1. A set $A$ with two binary operations $+$ and $\cdot$ is called a semiring if
   • $(A, +)$ is a commutative semigroup:
     • $(a + b) + c = a + (b + c)$ for all $a,b,c \in A$
     • $a + b = b + a$ for all $a,b \in A$
   • $(A, \cdot)$ is a semigroup:
     • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in A$
   • Multiplication left and right distributes over addition:
     • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all $a,b,c \in A$
     • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in A$

This is the "weakest" definition I've found and is from Udo Hebisch, Hanns J. Weinert: Semirings: Algebraic Theory and Application in Computer Science (German - Halbringe. Algebraische Theorie und Anwendungen in der Informatik) ISBN 3-519-02091-2

It does not require an identity element $0$ or an identity element 1 and does not require $0$ to be absorbing.

What would be an example of that definition?

2. A set $A$ with two binary operations $+$ and $\cdot$ is called a semiring if
   • $(A, +)$ is a commutative monoid:
     • $(a + b) + c = a + (b + c)$ for all $a,b,c \in A$
     • $0 + a = a + 0 = a$ for all $a \in A$
     • $a + b = b + a$ for all $a,b \in A$
   • $(A, \cdot)$ is a semigroup:
     • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in A$
   • Multiplication left and right distributes over addition:
     • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all $a,b,c \in A$
     • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in A$

This is basically the next step by adding an identity element $0$ to the requirements which expands the commutative semigroup $(A, +)$ to a commutative monoid. I've found this definition in my lecture notes from my algebra course.

What would be an example of that definition?

3. A set $A$ with two binary operations $+$ and $\cdot$ is called a semiring if
   • $(A, +)$ is a commutative monoid:
     • $(a + b) + c = a + (b + c)$ for all $a,b,c \in A$
     • $0 + a = a + 0 = a$ for all $a \in A$
     • $a + b = b + a$ for all $a,b \in A$
   • $(A, \cdot)$ is a monoid:
     • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in A$
     • $1 \cdot a = a \cdot 1 = a$ for all $a \in A$
   • Multiplication left and right distributes over addition:
     • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all $a,b,c \in A$
     • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in A$

This is also a definition I've seen which adds the requirement for an identity element $1$. This leads to $(A, \cdot)$ also being a monoid (not commutative though). I don't even remember if I've seen that definition somewhere or if I made it up but it's most probably from the book mentioned above.

What would be an example of that definition?

4. A set $A$ with two binary operations $+$ and $\cdot$ is called a semiring if
   • $(A, +)$ is a commutative monoid:
     • $(a + b) + c = a + (b + c)$ for all $a,b,c \in A$
     • $0 + a = a + 0 = a$ for all $a \in A$
     • $a + b = b + a$ for all $a,b \in A$
   • $(A, \cdot)$ is a semigroup:
     • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in A$
   • Multiplication left and right distributes over addition:
     • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all $a,b,c \in A$
     • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in A$
   • Multiplication by 0 annihilates R:
     • 0⋅a = a⋅0 = 0

This definition ditches the identity element 1 again reducing $(A, \cdot)$ back to semigroup again. This is the first time I have seen the requirement for $0$ to be absorbing which makes sense because $0a = 0$ does not hold in semigroups. I've found this definition also in the book mentioned above where it is called a "Hemiring".

What would be an example of that definition?

5. A set $A$ with two binary operations $+$ and $\cdot$ is called a semiring if
   • $(A, +)$ is a commutative monoid:
     • $(a + b) + c = a + (b + c)$ for all $a,b,c \in A$
     • $0 + a = a + 0 = a$ for all $a \in A$
     • $a + b = b + a$ for all $a,b \in A$
   • $(A, \cdot)$ is a monoid:
     • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in A$
     • $1 \cdot a = a \cdot 1 = a$ for all $a \in A$
   • Multiplication left and right distributes over addition:
     • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all $a,b,c \in A$
     • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in A$
   • Multiplication by 0 annihilates R:
     • 0⋅a = a⋅0 = 0

Basically the same definition as in $3.$ but with the requirement that $0$ is absorbing. This also makes sense as $0a = 0$ also does not hold in general monoids. I've found this definition on the English Wikipedia site which was by far the "strongest" definition in terms of requirements.

What would be an example of that definition?

I am not looking for the true definition as there is no such thing for semirings but if there are so many definitions with different requirements there have to be examples for each of them.

So, I am basically looking for examples which only fulfill the requirements of the stated definitions and not more because what would be the point in defining a semiring in a specific way then?

Answer 1

Example for 1): $(\{n\in\mathbb N|n>0\},+,\cdot)$

Example for 2): $(\mathbb N,+,\cdot)$

In 3), you said the multiplicative monoid isn't necessarily commutative, and yet you wrote the axiom. I'll assume you forgot to remove it. It also seems you explicitly want something with a nonabsorbing $0$. Example for 3): The best example I know of with nonabsorbing zero is the one I gave here, but it is also commutative. I'm afraid I can't do any better than that at the moment.

Example for 4): You could just use $(\{n\in\mathbb N|n\neq 1\},+,\cdot)$

Example for 5): $\mathbb N\langle x, y\rangle$, a polynomial semiring with noncommuting variables $x,y$, with coefficients from $\mathbb N$.

The literature for this topic is a bit of a mess. People use these structures in a lot of different ways. In my experience the most common definition requires an absorbing identity for $+$ and an identity for $\cdot$.

The resources I have used the most on this topic are

Gondran, M., & Minoux, M. (2008). Graphs, dioids and semirings: new models and algorithms (Vol. 41). Springer Science & Business Media.

Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.

The former is especially detailed about combinations of axioms.