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Modal operators on bounded commutative residuated \(\ell \)-monoids. (English) Zbl 1150.06016
Bounded commutative R\(\ell \)-monoids are bounded integral commutative residuated lattices satisfying the divisibility identity \(x\odot (x\rightarrow y)=x\wedge y\). The present paper generalizes results of {D. S. Macnab} [Algebra Univers. 12, 5–29 (1981; Zbl 0459.06005)]. A modal operator on an R\(\ell \)-monoid \(M\) is defined as a mapping \(f\colon M\to M\) such that (i) \(x\leq f(x)\), (ii) \(f(f(x))=f(x)\), (iii) \(f(x\odot y)=f(x)\odot f(y)\), and various properties of such mappings are studied.

MSC:
06F05 Ordered semigroups and monoids
06D35 MV-algebras
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