Differences

—		lattice-ordered_rings [2010/07/29 15:46] (current)
Line 1:		Line 1:
		+	=====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>
		+	)]