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Convergence with a fixed regulator in residuated lattices. (English) Zbl 1229.03058

The author investigates what the title says: convergence with a fixed regulator on residuated lattices. For example, the following results are proved:
(1)
Every sequence of an Archimedean residuated lattice has a unique \(v\)-limit. (Proposition 3.7)
(2)
If a residuated lattice \(L\) is complete, then it is \(v\)-Cauchy complete. (Theorem 3.17)

The notion of convergence with a fixed regulator is defined as follows. For a residuated lattice \({\mathcal L}=(L, \wedge, \vee, \odot, \to, 0,1)\) and \(v\in L\) such that \(v= v\odot v\), a sequence \(\{ x_n \}_n\) is said to be \(v\)-convergent to \(x\in L\) (or \(x\) is \(v\)-limit of \(\{ x_n \}_n\), denoted by \(x_n \to x\)) if for every \(p\in N\) there is \(n_0\in N\) such that \(d(x_n, x)^P \geq v\) for all \(n\in N\), \(n\geq n_0\), where \(d(x,y) = (x\to y) \wedge (y\to x)\). The element \(v\) is called a convergent regulator in \(L\).
A sequence \(\{ x_n \}_n\) is said to be \(v\)-Cauchy if for every \(p\in N\) there is \(n_0 \in N\) such that \(d(x_n , x_m)^p \geq v\) for all \(n,m \in N\), \(m\geq n\geq n_0\). A residuated lattice \(L\) is called \(v\)-Cauchy complete if every \(v\)-Cauchy sequence in \(L\) is \(v\)-convergent.

MSC:

03G25 Other algebras related to logic
06F05 Ordered semigroups and monoids
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