# Differences

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abelian_lattice-ordered_groups [2010/07/28 19:25]
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abelian_lattice-ordered_groups [2011/07/14 04:15] (current)
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====Definition==== ====Definition====
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An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a
[[lattice-ordered group]] [[lattice-ordered group]]
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==Morphisms== ==Morphisms==
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Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a
homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$. homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$.
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====Definition==== ====Definition====
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An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a
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[[commutative residuated lattice]] [[commutative residuated lattice]]
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$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity $\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity
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$x\cdot(x\to e)=e$. $x\cdot(x\to e)=e$.
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====Examples==== ====Examples====
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$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra. $\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.
====Basic results==== ====Basic results====
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The lattice reducts of (abelian) $\ell$-groups are [[distributive lattices]]. The lattice reducts of (abelian) $\ell$-groups are [[distributive lattices]].
====Properties==== ====Properties====
- +^[[Classtype]]                      |variety |
-^[[Classtype                      ]] |variety | +^[[Equational theory]]              |decidable |
- +^[[Quasiequational theory]]         |decidable |
-^[[Equational theory              ]] |decidable | +^[[First-order theory]]             |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] |
- +^[[Locally finite]]                 |no |
-^[[Quasiequational theory         ]] |decidable | +^[[Residual size]]                  | |
- +^[[Congruence distributive]]        |yes (see [[lattices]]) |
-^[[First-order theory             ]] |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] | +^[[Congruence modular]]             |yes |
- +^[[Congruence n-permutable]]        |yes, $n=2$ (see [[groups]]) |
-^[[Locally finite                 ]] |no | +^[[Congruence regular]]             |yes, (see [[groups]]) |
- +^[[Congruence uniform]]             |yes, (see [[groups]]) |
-^[[Residual size                  ]] | | +^[[Congruence extension property]]  | |
- +^[[Definable principal congruences]] | |
-^[[Congruence distributive        ]] |yes (see [[lattices]]) | +^[[Equationally def. pr. cong.]]    | |
- +^[[Amalgamation property]]          |yes |
-^[[Congruence modular             ]] |yes | +^[[Strong amalgamation property]]   |no [(CherriPowell1993)]
- +^[[Epimorphisms are surjective]]    | |
-^[[Congruence n-permutable        ]] |yes, $n=2$ (see [[groups]]) | +
- +
-^[[Congruence regular             ]] |yes, (see [[groups]]) | +
- +
-^[[Congruence uniform             ]] |yes, (see [[groups]]) | +
- +
-^[[Congruence extension property  ]] | | +
- +
-^[[Definable principal congruences ]] | | +
- +
-^[[Equationally def. pr. cong.    ]] | | +
- +
-^[[Amalgamation property          ]] |yes | +
- +
-^[[Strong amalgamation property   ]] | | +
- +
-^[[Epimorphisms are surjective    ]] | |+
====Finite members==== ====Finite members====
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None None
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-
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====Subclasses==== ====Subclasses====
- +[[Ordered abelian groups|Totally ordered abelian groups]]
-[[Totally ordered abelian groups]]  +
- +
- +
====Superclasses==== ====Superclasses====
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[[Representable lattice-ordered groups]] [[Representable lattice-ordered groups]]
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-
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====References==== ====References====
+[(Burris1985>
+Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups},
+Algebra Universalis,
+\textbf{20}, 1985, 400--401, http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/HerUndecLOAG.pdf)]
+[(CherriPowell1993>
+Mona Cherri and Wayne B. Powell,
+\emph{Strong amalgamation of lattice ordered groups and modules},
+International J. Math. & Math. Sci., Vol 16, No 1 (1993) 75--80, http://www.hindawi.com/journals/ijmms/1993/405126/abs/ doi:10.1155/S0161171293000080)]
[(Gurevic1967> [(Gurevic1967>
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Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},
-
Algebra i Logika Sem., Algebra i Logika Sem.,
- +\textbf{6}, 1967, 45--62)]
-\textbf{6}, 1967, 45--62 [[MRreview]])] +
- +
- +
- +
-[(Burris1985> +
- +
-Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups}, +
- +
-Algebra Universalis, +
- +
-\textbf{20}, 1985, 400--401 [[MRreview]])] +
- +
- +
- +
- +