Search

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest ∣G∣? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the idea of Song, Zhao and Huang [Czechoslov. Math. J., 70, 483–504 (2020)] and introduce the notion of right (left) Gorenstein subcategory r G (C , D) (l G (C , D)) , relative to two additive full subcategories C and D of an abelian category A . Under the assumption that C ⊆ D , we prove that the right Gorenstein subcategory r G (C , D) possesses many nice properties that it is closed under extensions, kernels of epimorphisms and direct summands. When C ⊆ D and C ⊥ D , we show that the right Gorenstein subcategory r G (C , D) admits some kind of stability. Then we discuss a resolution dimension for an object in A , called r G (C , D) -projective dimension. Finally, we prove that if (U , V) is a hereditary cotorsion pair with kernel C has enough injectives, such that U ⊆ D and U ⊥ D , then (r G (C , D) , r e s C ^ , C) is a weak Auslander—Buchweitz context, where res C ^ is the subcategory of A consisting of objects with finite C -projective dimension. [ABSTRACT FROM AUTHOR]

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng's paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)]. [ABSTRACT FROM AUTHOR]