Semirings with identity and zero

Abbreviation: SRng₀₁

Definition

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.

Morphisms

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.

Some results

Examples

Properties

Classtype	variety
Equational theory	decidable
Quasiequational theory
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	no
Congruence modular	no
Congruence n-permutable
Congruence regular
Congruence uniform
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?:
[Size 4]?:
[Size 5]?:
[Size 6]?:

Subclasses

Idempotent semirings with identity and zero
Rings with identity

Superclasses

Semirings with zero
Semirings with identity