Previous |  Up |  Next

Article

Keywords:
natural duality; dualisability; endodualisability; entailment; retraction
Summary:
A number of new results that say how to transfer the entailment relation between two different finite generators of a quasi-variety of algebras is presented. As their consequence, a well-known result saying that dualisability of a quasi-variety is independent of the generating algebra is derived. The transferral of endodualisability is also considered and the results are illustrated by examples.
References:
[1] Clark, D. M., Davey, B. A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998). MR 1663208 | Zbl 0910.08001
[2] Clark, D. M., Krauss, P. H.: Topological quasivarieties. Acta Sci. Math. (Szeged) 47 (1984), 3-39. MR 0755560
[3] Davey, B. A.: Dualisability in general and endodualisability in particular, Logic and Algebra (A. Ursini and P. Aglianò, eds.). Lecture Notes in Pure and Applied Mathematics, 389, Marcel Dekker, New York 437-455 (1996). MR 1404951
[4] Davey, B. A.: Personal communication.
[5] Davey, B. A., Haviar, M.: A schizophrenic operation which aids the efficient transfer of strong dualitites. Houston Math. J. 26 (2000), 215-222. MR 1814235
[6] Davey, B. A., Haviar, M.: Transferring optimal dualities: theory and practice. J. Austral. Math. Soc. 74 (2003), 393-420. DOI 10.1017/S1446788700003384 | MR 1970056 | Zbl 1047.08006
[7] Davey, B. A., Haviar, M., Priestley, H. A.: The syntax and semantics of entailment in duality theory. J. Symbolic Logic 60 (1995), 1087-1114. DOI 10.2307/2275875 | MR 1367197 | Zbl 0845.08006
[8] Davey, B. A., Haviar, M., Willard, R.: Structural entailment. Algebra Universalis 54 (2005), 397-416. DOI 10.1007/s00012-005-1944-y | MR 2218853 | Zbl 1090.08009
[9] Davey, B. A., Pitkethly, J. G.: Endoprimal algebras. Algebra Universalis 38 (1997), 266-288. DOI 10.1007/s000120050055 | MR 1619762 | Zbl 0934.08004
[10] Davey, B. A., Werner, H.: Dualities and equivalences for varieties of algebras. Contributions to Lattice Theory (Szeged, 1980), (A. Huhn and E. T. Schmidt, eds.), Coll. Math. Soc. János Bolyai 33, North-Holland, Amsterdam (1983), 101-275. MR 0724265 | Zbl 0532.08003
[11] Davey, B. A., Willard, R.: The dualisability of a quasi-variety is independent of the generating algebra. Algebra Universalis 45 (2001), 103-106. DOI 10.1007/s000120050204 | MR 1809859 | Zbl 1039.08006
[12] Pitkethly, J. G., Davey, B. A.: Dualisability: Unary Algebras and Beyond. Springer (2005). MR 2161626 | Zbl 1085.08001
[13] Pontryagin, L. S.: Sur les groupes abéliens continus. C.R. Acad. Sci. Paris 198 (1934), 238-240.
[14] Pontryagin, L. S.: The theory of topological commutative groups. Ann. Math. 35 (1934), 361-388. DOI 10.2307/1968438 | MR 1503168 | Zbl 0009.15601
[15] Priestley, H. A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2 (1970), 186-190. DOI 10.1112/blms/2.2.186 | MR 0265242 | Zbl 0201.01802
[16] Priestley, H. A.: Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. 24 (1972), 507-530. MR 0300949 | Zbl 0323.06011
[17] Saramago, M.: A study of natural dualities, including an analysis of the structure of failsets. Ph.D. thesis, University of Lisbon (1998).
[18] Saramago, M.: Some remarks on dualisability and endodualisability. Algebra Universalis 43 (2000), 197-212. DOI 10.1007/s000120050153 | MR 1773938 | Zbl 1011.08003
[19] Saramago, M. J., Priestley, H. A.: Optimal natural dualities: the structure of failsets. Internat. J. Algebra Comput. 12 (2002), 407-436. DOI 10.1142/S0218196702000791 | MR 1910686 | Zbl 1027.08006
[20] Stone, M. H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 4 (1936), 37-111. MR 1501865 | Zbl 0014.34002
[21] Wegener, C. B.: Natural dualities for varieties generated by lattice-structured algebras. Ph.D. thesis, University of Oxford (1999).
[22] Willard, R.: New tools for proving dualizability. In: Dualities, Interpretability and Ordered Structures (Lisbon, 1997) (J. Vaz de Carvalho and I. Ferreirim, eds.), Centro de Álgebra da Universidade de Lisboa (1999), 69-74.
[23] Zádori, L.: Natural duality via a finite set of relations. Bull. Austral. Math. Soc. 51 (1995), 469-478. DOI 10.1017/S0004972700014301 | MR 1331440
Partner of
EuDML logo