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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)]