This is an old revision of the document!

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