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idempotent_semirings_with_identity [2010/07/29 15:46] (current)
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 +=====Idempotent semirings with identity=====
 +Abbreviation: **ISRng$_1$**
 +
 +====Definition====
 +An \emph{idempotent semiring with identity} is a [[semirings with identity]] $\mathbf{S}=\langle S,\vee,\cdot,1
 +\rangle $ such that
 +
 +$\vee$ is idempotent:  $x\vee x=x$
 +
 +==Morphisms==
 +Let $\mathbf{S}$ and $\mathbf{T}$ be idempotent semirings with identity. A morphism from $\mathbf{S}$
 +to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
 +
 +====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====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Idempotent semirings with identity and zero]]
 +
 +====Superclasses====
 +[[Idempotent semirings]]
 +
 +[[Semirings with identity]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]