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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>
 +)]