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Retract varieties of lattice ordered groups. (English) Zbl 0705.06011

D. Duffus and I. Rival have introduced the notion of an order variety as a class of partially ordered sets closed under the formation of direct products and retracts [cf. Discrete Math. 35, 53-118 (1981; Zbl 0459.06002)]. In an analogous way, a class of lattice-ordered groups is called a retract variety if it is closed under the formation of direct products and (lattice-group) retracts. The author shows that a retract G of a direct product H of lattice-ordered groups \(H_ i\) is isomorphic to a direct product of retracts \(G_ i\) of the \(H_ i\). As a consequence, the class \({\mathcal R}\) of retract varieties of lattice-ordered groups is very large. Ordered by inclusion it is a complete Brouwerian lattice, it has a lot of atoms, but no co-atom.
Reviewer: K.Keimel

MSC:

06F15 Ordered groups
08B15 Lattices of varieties

Citations:

Zbl 0459.06002
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References:

[1] D. Duffus W. Poguntke I. Rival: Retracts and the fixed point problem for finite partially ordered sets. Canad. Math. Bull. 23, 1980, 231 - 236. · Zbl 0441.06003 · doi:10.4153/CMB-1980-031-2
[2] D. Duffus I. Rival: Retracts of partially ordered sets. J. Austral. Math. Soc., Ser. A, 27, 1979, 495-506. · Zbl 0421.06002 · doi:10.1017/S1446788700013483
[3] D. Duffus I. Rival M. Simonovits: Spanning retracts of a partially ordered set. Discrete Math. 32, 1980, 1-7. · Zbl 0453.06003 · doi:10.1016/0012-365X(80)90093-X
[4] D. Duffus I. Rival: A structure theory for ordered sets. Discrete Math. 35, 1981, 53-118. · Zbl 0459.06002 · doi:10.1016/0012-365X(81)90201-6
[5] J. Jakubík: Retracts of abelian lattice ordered groups. Czechoslov. Math. J. 39, 1989, 477-485. · Zbl 0691.06009
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