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:
| [1] | T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer Verlag, London, 2005. · Zbl 1073.06001 |
| [2] | T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press, 1972. · Zbl 0301.06001 |
| [3] | E. D. Foreman, Order automorphisms on the lattice of residuated maps of some special nondistributive lattices, University of Louisville, Electronic Theses and Dissertations, Paper 2257, 2015. DOI · doi:10.18297/etd/2257 |
| [4] | G. Grätzer and E. T. Schmidt, On the lattice of all join-endomorphisms of a lattice, Proc. Amer. Math. Soc. 9 (1958), 722-726. · Zbl 0087.26104 |
| [5] | M. F. Janowitz, Tolerances and congruences on lattices, Czechoslovak Math. J. 36 (1986), 108-115. · Zbl 0598.06004 |
| [6] | K. Kaarli and V. Kuchmei, Order affine completeness of lattices with boolean congru-ence lattices, Czechoslovak Math. J. 57 (2007), 1049-1065. · Zbl 1174.06304 |
| [7] | K. Kaarli, V. Kuchmei, and S. E. Schmidt, Sublattices of the direct product, Algebra Universalis 59 (2008), 85-95. · Zbl 1170.06003 |
| [8] | K. Kaarli and S. Radeleczki, Representation of integral quantales by tolerances, Alge-bra Universalis 79 (2018), Paper No. 5, 20 pp. DOI · Zbl 1471.06009 · doi:10.1007/s00012-018-0484-1 |
| [9] | K. Kaarli and S. Radeleczki, Ordered relations, concept lattices and self-bonds, J. Multiple Valued Log. Soft Comput. 36 (2021), 321-336. · Zbl 1529.06006 |
| [10] | V. Kuchmei and S. E. Schmidt, Minimal residuated maps on lattices of finite projective geometries, Manuscript. |
| [11] | G. N. Raney, Tight Galois connections and complete distributivity, Trans. Amer. Math. Soc. 97 (1960), 418-426. · Zbl 0098.02703 |
| [12] | E. A. Schreiner, Tight residuated mappings, Proc. Univ. Houston (Lattice Theory Conference, Houston 1973), 520-530. · Zbl 0296.06007 |
| [13] | Z. Shmuely, The structure of Galois connections, Pacific J. Math. 54 (1974), 209-225. · Zbl 0275.06003 |
| [14] | S. Valentini, Representation theorems for quantales, Math. Logic Quart. 40 (1994) 182-190. · Zbl 0816.06018 |
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