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abelian_lattice-ordered_groups [2010/07/28 19:20]
127.0.0.1 external edit
abelian_lattice-ordered_groups [2011/07/14 04:15] (current)
jipsen
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-=====Abelian lattice-ordered groups=====  +=====Abelian lattice-ordered groups=====
-  +
-Abbreviation: **AbLGrp**  +Abbreviation: **AbLGrp**
-  +
-====Definition====  +
-An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a  +====Definition====
-[[lattice-ordered group]]  +An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a
-$\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that  +[[lattice-ordered group]]
-  +$\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that
-$\cdot$ is commutative:  $x\cdot y=y\cdot x$  +
-  +$\cdot$ is commutative:  $x\cdot y=y\cdot x$
-==Morphisms==  +
-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  +==Morphisms==
-homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$.  +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 $f(x\cdot y)=f(x)\cdot f(y)$.
-Remark: It follows that $f(x\wedge y)=f(x)\wedge f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$  +
-  +Remark: It follows that $f(x\wedge y)=f(x)\wedge f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$
-====Definition====  +
-An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a  +
-[[commutative residuated lattice]]  +====Definition====
-$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity  +An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a
-$x\cdot(x\to e)=e$.  +[[commutative residuated lattice]]
-  +$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity
-Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$  +$x\cdot(x\to e)=e$.
-  +
-\begin{examples}  +Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$
-$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.  +
-\end{examples}  +
-  +====Examples====
-\begin{basic_results}  +$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.
-The lattice reducts of (abelian) $\ell$-groups are [[distributive lattices]].  +
-\end{basic_results}  +
-  +====Basic results====
-====Properties====  +The lattice reducts of (abelian) $\ell$-groups are [[distributive lattices]].
-^[[Classtype                      ]] |variety |  +
-^[[Equational theory              ]] |decidable |  +
-^[[Quasiequational theory         ]] |decidable |  +====Properties====
-^[[First-order theory             ]] |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] |  +^[[Classtype]]                      |variety |
-^[[Locally finite                 ]] |no |  +^[[Equational theory]]              |decidable |
-^[[Residual size                  ]] | |  +^[[Quasiequational theory]]         |decidable |
-^[[Congruence distributive        ]] |yes (see [[lattices]]) |  +^[[First-order theory]]             |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] |
-^[[Congruence modular             ]] |yes |  +^[[Locally finite]]                 |no |
-^[[Congruence n-permutable        ]] |yes, $n=2$ (see [[groups]]) |  +^[[Residual size]]                  | |
-^[[Congruence regular             ]] |yes, (see [[groups]]) |  +^[[Congruence distributive]]        |yes (see [[lattices]]) |
-^[[Congruence uniform             ]] |yes, (see [[groups]]) |  +^[[Congruence modular]]             |yes |
-^[[Congruence extension property  ]] | |  +^[[Congruence n-permutable]]        |yes, $n=2$ (see [[groups]]) |
-^[[Definable principal congruences ]] | |  +^[[Congruence regular]]             |yes, (see [[groups]]) |
-^[[Equationally def. pr. cong.    ]] | |  +^[[Congruence uniform]]             |yes, (see [[groups]]) |
-^[[Amalgamation property          ]] |yes |  +^[[Congruence extension property]]  | |
-^[[Strong amalgamation property   ]] | |  +^[[Definable principal congruences]] | |
-^[[Epimorphisms are surjective    ]] | |  +^[[Equationally def. pr. cong.]]    | |
- +^[[Amalgamation property]]          |yes |
+^[[Strong amalgamation property]]   |no [(CherriPowell1993)]
+^[[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====  +[(Burris1985
-  +Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups},
-[(Gurevic1967>  +Algebra Universalis
-Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},  +\textbf{20}, 1985, 400--401, http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/HerUndecLOAG.pdf)
-Algebra i Logika Sem.,  +
-\textbf{6}, 1967, 45--62 [[MRreview]]  +[(CherriPowell1993
-  +Mona Cherri and Wayne B. Powell,
-[(Burris1985>  +\emph{Strong amalgamation of lattice ordered groups and modules},
-Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups},  +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)]
-Algebra Universalis,  +
-\textbf{20}, 1985, 400--401 [[MRreview]]  +[(Gurevic1967>
-)]  +Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},
-  +Algebra i Logika Sem.
- +\textbf{6}, 1967, 45--62)]