A semiring with identity and zero is a structure S = (S, + ,0,·,1
) of type (2,0,2,0) such that
(S, + ,0) is a commutative monoid
(S,·,1) is a monoid
0 is a zero for ·: 0·x = 0 and x·0 = 0
· distributes over + : x·(y + z) = x·y + x·z and (y + z)·x = y·x + z·x.
Let S and T be semirings with identity and zero. A morphism from S to T is a function h : S→T that is a homomorphism: h(x + y) = h(x) + h(y) and h(x·y) = h(x)·h(y) and h(0) = 0 and h(1) = 1.
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|