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On general primarities which are equivalent to primality and tertiarity. (Russian) Zbl 0628.06008
Let L and R be complete lattices. L is called a lattice of operators of R if for any \(a\in R\) and \(x\in L\) the product xa\(\in R\) is defined, such that xa\(\leq a\), \(x(a+b)=xa+ya\), \((x+y)a=xa+xb\) (where \(b\in R\), \(y\in L)\). For a modular lattice R (which satisfies some additional conditions), primarity is a map \(s: R\to L\), such that \(s(1)=1\), and for any \(a\in R\), \(a\neq 1\), \(r(a,s(a))>a\), where \(r(a,x)=\sup \{b:\) \(b\in R\), xb\(\leq a\}\). The author proves equivalence criteria to some special kind of primarity, which are useful for classification of primarities.
Reviewer: V.Meskhi
MSC:
06C05 Modular lattices, Desarguesian lattices
06B23 Complete lattices, completions
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