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Finitely presented lattice-ordered abelian groups with order-unit. (English) Zbl 1261.06022

Let \((G, u)\) be a unital \(l\)-group. A basis of \(G\) is a set \(B = \{b_1, \cdots , b_n \}\) of strictly positive elements such that
(i) \(\langle B\rangle=G\);
(ii) there are natural numbers \(m_1, \ldots ,m_n\) such that \(u=\sum_{i+1}^{n}m_ib_i;\)
(iii) for each \(k = 1, 2, \ldots\) and \(k\)-element subset \(C\) of \(B\) with \(0\neq \bigwedge \{b \mid b \in C\}\), the set \(\{ m \in \text{MaxSpec}(G) \mid m \supseteq B\setminus C\}\) is homeomorphic to a \((k -1)\)-simplex, where MaxSpec(\(G\)) denotes the set of maximal ideals of \(G\) with the spectral topology.
The authors show that a unital \(l\)-group is finitely presented iff it has a basis, which extends the Baker-Beynon theorem of abelian \(l\)-groups. Furthermore, a large class of projectives is constructed from bases having special properties.
The reviewer would like to recommend this nice paper to readers in the field of \(l\)-groups and MV-algebras.

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
08B30 Injectives, projectives
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References:

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