# Differences

This shows you the differences between two versions of the page.

abelian_lattice-ordered_groups [2010/07/28 20:26] jipsen |
abelian_lattice-ordered_groups [2011/07/14 04:15] (current) jipsen |
||
---|---|---|---|

Line 5: | Line 5: | ||

====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]] | ||

Line 13: | Line 12: | ||

==Morphisms== | ==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 | 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 $f(x\cdot y)=f(x)\cdot f(y)$. | ||

Line 21: | Line 19: | ||

====Definition==== | ====Definition==== | ||

- | |||

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]] | ||

Line 31: | Line 28: | ||

====Examples==== | ====Examples==== | ||

- | |||

$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, 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==== | ||

- | |||

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 | |

- | ^[[Equational theory]] |decidable | | + | ^[[Quasiequational theory]] |decidable | |

- | ^[[Quasiequational theory]] |decidable | | + | ^[[First-order theory]] |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] | |

- | ^[[First-order theory]] |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] | | + | ^[[Locally finite]] |no | |

- | ^[[Locally finite]] |no | | + | ^[[Residual size]] | | |

- | ^[[Residual size]] | | | + | ^[[Congruence distributive]] |yes (see [[lattices]]) | |

- | ^[[Congruence distributive]] |yes (see [[lattices]]) | | + | ^[[Congruence modular]] |yes | |

- | ^[[Congruence modular]] |yes | | + | ^[[Congruence n-permutable]] |yes, $n=2$ (see [[groups]]) | |

- | ^[[Congruence n-permutable]] |yes, $n=2$ (see [[groups]]) | | + | ^[[Congruence regular]] |yes, (see [[groups]]) | |

- | ^[[Congruence regular]] |yes, (see [[groups]]) | | + | ^[[Congruence uniform]] |yes, (see [[groups]]) | |

- | ^[[Congruence uniform]] |yes, (see [[groups]]) | | + | ^[[Congruence extension property]] | | |

- | ^[[Congruence extension property]] | | | + | |

^[[Definable principal congruences]] | | | ^[[Definable principal congruences]] | | | ||

- | ^[[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==== | ||

+ | [(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> | ||

- | |||

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]])] | + | |

- | | + | |

- | | + | |

- | | + | |

- | | + | |

Trace: