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commutative_lattice-ordered_rings [2010/07/29 15:46] (current)
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+=====Commutative lattice-ordered rings=====
+
+Abbreviation: **CLRng**
+
+====Definition====
+A \emph{commutative lattice-ordered ring} is a [[lattice-ordered ring]] $\mathbf{A}=\langle A,\vee,\wedge,+,-,0,\cdot\rangle$ such that
+
+$\cdot$ is \emph{commutative}:  $xy=yx$
+
+Remark: This is a template.
+If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.
+
+It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be commutative lattice-ordered rings. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+$h(x \vee y)=h(x) \vee h(y)$,
+$h(x \wedge y)=h(x) \wedge h(y)$,
+$h(x + y)=h(x) + h(y)$,
+$h(x \cdot y)=h(x) \cdot h(y)$.
+
+====Definition====
+A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +...\rangle$ such that
+
+$...$ is ...:  $axiom$
+
+$...$ is ...:  $axiom$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
+
+^[[Classtype]]                        |(value, see description) [(Ln19xx)]  |
+^[[Equational theory]]                | |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               | |
+^[[Locally finite]]                   | |
+^[[Residual size]]                    | |
+^[[Congruence distributive]]          |yes  |
+^[[Congruence modular]]               |yes  |
+^[[Congruence $n$-permutable]]        |yes, $n=2$ |
+^[[Congruence regular]]               |yes  |
+^[[Congruence uniform]]               |yes  |
+^[[Congruence extension property]]    | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]      | |
+^[[Amalgamation property]]            | |
+^[[Strong amalgamation property]]     | |
+^[[Epimorphisms are surjective]]      | |
+
+====Finite members====
+
+====Subclasses====
+  [[Commutative f-rings]] subvariety
+
+
+====Superclasses====
+  [[Lattice-ordered rings]] supervariety
+
+  [[Abelian lattice-ordered groups]] subreduct
+
+  [[Commutative rings]] subreduct
+
+
+====References====
+
+[(Lastname19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]
+
+

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