# Differences

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lattice-ordered_rings [2010/07/29 15:46] (current)
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+=====Lattice-ordered rings=====
+Abbreviation: **LRng**
+====Definition====
+A \emph{lattice-ordered ring} (or $\ell$\emph{-ring}) is a structure $\mathbf{L}=\langle L,\vee,\wedge,+,-,0,\cdot\rangle$ such that
+
+
+$\langle L,\vee,\wedge\rangle$ is a [[lattice]]
+
+
+$\langle L,+,-,0,\cdot\rangle$ is a [[ring]]
+
+
+$+$ is order-preserving:  $x\leq y\Longrightarrow x+z\leq y+z$
+
+
+${\uparrow}0$ is closed under $\cdot$:  $0\leq x,y\Longrightarrow 0\leq x\cdot y$
+
+
+Remark:
+
+====Definition====
+==Morphisms==
+Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-rings. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a
+homomorphism: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x+y)=f(x)+f(y)$.
+====Examples====
+
+
+====Basic results====
+The lattice reducts of lattice-ordered rings are [[distributive lattices]].
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  | |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Congruence distributive]]  |yes, see [[lattices]] |
+^[[Congruence extension property]]  | |
+^[[Congruence n-permutable]]  |yes, $n=2$, see [[groups]] |
+^[[Congruence regular]]  |yes, see [[groups]] |
+^[[Congruence uniform]]  |yes, see [[groups]] |
+
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  | |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+
+====Finite members====
+
+$\begin{array}{lr} +None +\end{array}$
+
+====Subclasses====
+[[Commutative lattice-ordered rings]]
+
+====Superclasses====
+[[Abelian lattice-ordered groups]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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