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The variety generated by order algebras. (English) Zbl 1058.08001

Every ordered set \((A,\leq )\) can be considered as an algebra \((A,\cdot )\) in a natural way, where \(xy=x\) iff \(x\leq y\) and \(xy=y\) otherwise. The variety generated by order algebras is investigated. It is proved, among other things, that this variety is not finitely based and, although locally finite, it is not contained in any finitely generated variety. Also, the bottom of the lattice of its subvarieties is described.

MSC:

08A05 Structure theory of algebraic structures
08B15 Lattices of varieties
06A11 Algebraic aspects of posets
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