# Article

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Keywords:
reflexive bilattice; hyperreflexive bilattice; subspace lattice; bilattice
Summary:
The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $\mathcal {L}$ we can construct the bilattice $\Sigma _{\mathcal {L}}$. Similarly, having a bilattice $\Sigma$ we may consider the lattice $\mathcal {L}_{\Sigma }$. In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples of hyperreflexive or not hyperreflexive bilattices are given.
References:
[1] Arveson, W.: Interpolation problems in nest algebras. J. Funct. Anal. 20 (1975), 208-233. DOI 10.1016/0022-1236(75)90041-5 | MR 0383098
[2] Davidson, K. R., Harrison, K. J.: Distance formulae for subspace lattices. J. Lond. Math. Soc., (2) 39 (1989), 309-323. DOI 10.1112/jlms/s2-39.2.309 | MR 0991664 | Zbl 0723.47003
[3] Kli{ś}-Garlicka, K.: Reflexivity of bilattices. Czech. Math. J. 63 (2013), 995-1000. DOI 10.1007/s10587-013-0067-4 | MR 3165510 | Zbl 1313.47024
[4] Kraus, J., Larson, D. R.: Reflexivity and distance formulae. Proc. Lond. Math. Soc. (3) 53 (1986), 340-356. DOI 10.1112/plms/s3-53.2.340 | MR 0850224
[5] Shulman, V., Turowska, L.: Operator synthesis. I. Synthetic sets, bilattices and tensor algebras. J. Funct. Anal. 209 (2004), 293-331. DOI 10.1016/S0022-1236(03)00270-2 | MR 2044225 | Zbl 1071.47066
[6] Shulman, V.: A review of "Nest Algebras by K. R. Davidson, Longman Sci. and Techn. Pitman Research Notes Math., 1988". Algebra and Analiz 2 (1990), 236-255 http://www.mathnet.ru/links/04e6653e78d90590f32a76de1b827b3b/aa194.pdf

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