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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>
+)]

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