semirings_with_identity - MathStructures

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=====Semirings with identity=====

Abbreviation: **SRng$_1$**

====Definition====
A \emph{semiring with identity} is a structure $\mathbf{S}=\langle S,+,\cdot,1
\rangle $ of type $\langle 2,2,0\rangle $ such that


$\langle S,+\rangle $ is a [[commutative semigroup]]


$\langle S,\cdot, 1\rangle$ is a [[monoid]]


$\cdot$ distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be semirings with zero. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

====Examples====
Example 1: 

====Basic results====

====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 def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &11\\
f(4)= &73\\
f(5)= &703\\
f(6)= &\\
\end{array}$

====Subclasses====
[[semirings with identity and zero]] 

[[semifields]]

====Superclasses====
[[Semirings]] 


====References====

[(Ln19xx>
)]

Trace: • commutative_monoids • compact_topological_spaces • abelian_partially_ordered_groups • continuous_posets • equational_theory • abelian_lattice-ordered_groups • t0-spaces • right_hoops • distributive_allegories • finite_commutative_semigroups