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abelian_lattice-ordered_groups [2010/07/28 19:11]
jipsen created
abelian_lattice-ordered_groups [2011/07/14 04:15] (current)
jipsen
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-Hello+=====Abelian lattice-ordered groups===== 
 + 
 +Abbreviation: **AbLGrp** 
 + 
 + 
 +====Definition==== 
 +An \emph{abelian lattice-ordered group} (or abelian $\ell $\emph{-group}) is a  
 +[[lattice-ordered group]] 
 +$\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that 
 + 
 +$\cdot$ 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 
 +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 $f(e)=e$ 
 + 
 + 
 +====Definition==== 
 +An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a  
 +[[commutative residuated lattice]]  
 +$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle $ that satisfies the identity  
 +$x\cdot(x\to e)=e$. 
 + 
 +Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$ 
 + 
 + 
 +====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. 
 + 
 + 
 +====Basic results==== 
 +The lattice reducts of (abelian) $\ell$-groups are [[distributive lattices]]. 
 + 
 + 
 +====Properties==== 
 +^[[Classtype]]                       |variety | 
 +^[[Equational theory]]               |decidable | 
 +^[[Quasiequational theory]]          |decidable | 
 +^[[First-order theory]]              |hereditarily undecidable [(Gurevic1967)] [(Burris1985)] | 
 +^[[Locally finite]]                  |no | 
 +^[[Residual size]]                   | | 
 +^[[Congruence distributive]]         |yes (see [[lattices]]) | 
 +^[[Congruence modular]]              |yes | 
 +^[[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]]    |no [(CherriPowell1993)] | 
 +^[[Epimorphisms are surjective]]     | | 
 + 
 + 
 +====Finite members==== 
 +None 
 + 
 + 
 +====Subclasses==== 
 +[[Ordered abelian groups|Totally ordered abelian groups]]  
 + 
 + 
 +====Superclasses==== 
 +[[Representable lattice-ordered groups]]  
 + 
 + 
 +====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> 
 +Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, 
 +Algebra i Logika Sem., 
 +\textbf{6}, 1967, 45--62)]