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abelian_lattice-ordered_groups [2010/07/29 10:53]
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abelian_lattice-ordered_groups [2011/07/14 04:15] (current)
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====Definition==== ====Definition====
- 
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
[[commutative residuated lattice]] [[commutative residuated lattice]]
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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====
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^[[Classtype]]                       |variety | ^[[Classtype]]                       |variety |
^[[Equational theory]]               |decidable | ^[[Equational theory]]               |decidable |
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^[[Equationally def. pr. cong.]]     | | ^[[Equationally def. pr. cong.]]     | |
^[[Amalgamation property]]           |yes | ^[[Amalgamation property]]           |yes |
-^[[Strong amalgamation property]]    | |+^[[Strong amalgamation property]]    |no [(CherriPowell1993)] |
^[[Epimorphisms are surjective]]     | | ^[[Epimorphisms are surjective]]     | |
====Finite members==== ====Finite members====
- 
None None
====Subclasses==== ====Subclasses====
- +[[Ordered abelian groups|Totally ordered abelian groups]]
-[[Totally ordered abelian groups]] +
====Superclasses==== ====Superclasses====
- 
[[Representable lattice-ordered groups]] [[Representable lattice-ordered groups]]
====References==== ====References====
-[(Gurevic1967> 
-Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, 
-Algebra i Logika Sem., 
-\textbf{6}, 1967, 45--62 [[MRreview]])] 
- 
[(Burris1985> [(Burris1985>
Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups}, Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups},
Algebra Universalis, Algebra Universalis,
-\textbf{20}, 1985, 400--401 [[MRreview]])]+\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> 
 +Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, 
 +Algebra i Logika Sem., 
 +\textbf{6}, 1967, 45--62)]