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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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