# Differences

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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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