idempotent_semirings_with_identity_and_zero

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=====Idempotent semirings with identity and zero=====

Abbreviation: **ISRng**$_{01}$

====Definition====
An \emph{idempotent semiring with identity and zero} is a [[semirings with identity and zero]] $\mathbf{S}=\langle S,\vee,0,\cdot,1
\rangle $ such that 
$\vee$ is idempotent:  $x\vee x=x$

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

====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 meet-semidistributive]]  |yes |
^[[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====

$\begin{array}{lr}
f(1)= &   1\\
f(2)= &   1\\
f(3)= &   3\\
f(4)= &  20\\
f(5)= & 149\\
f(6)= &1488\\
f(7)= &18554
\end{array}$

====Subclasses====
[[Kleene algebras]] 

====Superclasses====
[[Idempotent semirings with zero]] 

[[Idempotent semirings with identity]] 

[[Semirings with identity and zero]] 


====References====

[(Ln19xx>
)]

Trace: • sequential_algebras • reflexive_relations • pseudo_basic_logic_algebras • commutative_monoids • abelian_groups • residuated_partially_ordered_semigroups • weakly_representable_relation_algebras • sets • finite_commutative_binars • commutative_regular_rings