Atoms of the lattices of residuated mappings. (English) Zbl 1537.06001
Summary: Given a lattice \(L\), we denote by \(\mathrm{Res}(L)\) the lattice of all residuated maps on \(L\). The main objective of the paper is to study the atoms of \(\mathrm{Res}(L)\) where \(L\) is a complete lattice. Note that the description of dual atoms of \(\mathrm{Res}(L)\) easily follows from earlier results of
Z. Shmuely [Pac. J. Math. 54, No. 2, 209–225 (1974; Zbl 0275.06003)]. We first consider lattices \(L\) for which all atoms of \(\mathrm{Res}(L)\) are mappings with 2-element range and give a sufficient condition for this. Extending this result, we characterize these atoms of \(\mathrm{Res}(L)\) which are weakly regular residuated maps in the sense of
T. S. Blyth and M. F. Janowitz [Residuation theory. Oxford: Pergamon Press; Warszawa: PWN - Polish Scienctific Publishers, (1972; Zbl 0301.06001)]. In the rest of the paper we investigate the atoms of \(\mathrm{Res}(M)\) where \(M\) is the lattice of a finite projective plane, in particular, we describe the atoms of \(\mathrm{Res}(F)\), where \(F\) is the lattice of the Fano plane.
MSC:
06B23 | Complete lattices, completions |
06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
Keywords:
complete lattice; residuated mapping; minimal (maximal) residuated mapping; decreasing pair; weakly regular residuated mapping; tolerance simple lattice; projective plane; Fano plane; Fano latticeReferences:
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