Abstract
A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued relations.
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References
Bandelt, H.-J.: Regularity and complete distributivity. Semigroup Forum 19, 123–126 (1980)
Erceg, M.A.: Functions, equivalence relations, quotient spaces and subsets in fuzzy set theory. Fuzzy Sets Syst. 3, 75–92 (1980)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin, Heidelberg, New York (1980)
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York, Heidelberg, Berlin (1971)
Markowsky, G.: Idempotents and product representations with applications to semigroup of binary relations. Semigroup Forum 5, 95–119 (1972)
Mulvey, C.J., Pelletier, J.W.: A quantisation of the calculus of relations. In: CMS Proceedings, vol. 13, pp. 345–360. Am. Math. Soc., Providence (1992)
Pelletier, J.W., Rosický, J.: Simple involutive quantales. J. Algebra 195, 367–386 (1997)
Plemmons, R.J., West, M.T.: On the semigroup of binary relations. Pac. J. Math. 35, 743–753 (1970)
Raney, G.N.: A subdirect-union representation of completely distributive lattices. Proc. Am. Math. Soc. 4, 518–522 (1953)
Rosenthal, K.I.: Quantales and Their Applications. Pitman Research Notes in Mathematics, vol. 348. Longman Scientific & Technical, Longman House, Burnt Mill, Harlow (1996)
Schein, B.M.: Regular elements of the semigroup of all binary relations. Semigroup Forum 13, 95–112 (1976)
Xu, X.-Q., Luo, M.-K.: Regular relations and normality of topologies. Semigroup Forum 72, 477–480 (2006)
Yang, J.C.: A theorem on the semigroup of binary relations. Proc. Am. Math. Soc. 22, 134–135 (1969)
Zareckiĭ, K.A.: The semigroup of binary relations. Mat. Sb. 61, 291–305 (1963) (in Russian)
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Communicated by Boris M. Schein.
The second named author gratefully acknowledges the grant MTM2009-12872-C02-02 from the Ministry of Science and Innovation of Spain.
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Höhle, U., Kubiak, T. On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem. Semigroup Forum 83, 313–319 (2011). https://doi.org/10.1007/s00233-011-9311-0
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DOI: https://doi.org/10.1007/s00233-011-9311-0