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Keywords:
concrete category; optimal subset; reflexive subobject lattice; reflexive endomorphism algebra
concrete category; optimal subset; reflexive subobject lattice; reflexive endomorphism algebra
Summary:
In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category {\bf Set} of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.
References:
[1] Davey B.A., Priestley H.A.: Introduction to Lattices and Order. Cambridge Text Book, Cambridge University Press, 1994. MR 1902334 | Zbl 1002.06001
[2] Halmos P.R.: Ten problems in Hilbert space. Bull. Amer. Math. Soc. 76 (1970), 887-933. MR 0270173 | Zbl 0204.15001
[3] Halmos P.R.: Reflexive lattices of subspaces. J. London Math. Soc. 4 (1971), 257-263. MR 0288612 | Zbl 0231.47003
[4] Preuss G.: Theory of Topological Structures - An approach to Categorical Topology. D. Reidel Publishing Company, 1988. MR 0937052 | Zbl 0649.54001
[5] Harrison K.J., Longstaff W.E.: Automorphic images of commutative subspace lattices. Proc. Amer. Math. Soc. 296 (1) (1986), 217-228. MR 0837808 | Zbl 0616.46021
[6] Longstaff W.E.: Strongly reflexive subspace lattices. J. London Math. Soc. (2) 11 (1975), 491-498. MR 0394233
[7] Longstaff W.E.: On lattices whose every realization on Hilbert space is reflexive. J. London Math. Soc. (2) 37 (1988), 499-508. MR 0939125 | Zbl 0654.47025
[8] Longstaff W.E., Oreste P.: On the ranks of single elements of reflexive operator algebras. Proc. Amer. Math. Soc. 125 (10) (1997), 2875-2882. MR 1402872 | Zbl 0883.47024
[9] Manes E.G.: Algebraic Theories. Graduate Texts in Mathematics 26, Springer-Verlag, 1976. MR 0419557 | Zbl 0489.18003
[10] Ore O.: Structures and group theory I. Duke Math. J. 3 (1937), 149-173. MR 1545977 | Zbl 0016.35103
[11] Ore O.: Structures and group theory II. Duke Math. J. 4 (1938), 247-269. MR 1546048 | Zbl 0020.34801
[12] Schmidt R.: Subgroup Lattices of Groups. De Gruyter Expositions in Mathematics 14, Walter de Gruyter, Berlin-New York, 1994. MR 1292462 | Zbl 1026.20015
[13] Raney G.N.: Completely distributive lattices. Proc. Amer. Math. Soc. 3 (1952), 677-680. MR 0052392
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