Abstract
In 1999, for lattices A and B, G. Grtzer and F. Wehrung introduced the lattice tensor product, A⊠B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A⊠B.
Abstract
G. Grtzer and F. Wehrung has recently introduced the lattice tensor product, A⊠B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A⊠B, obtaining the isomorphism Id(A⊠B)≌A⊠Id B. This generalizes an earlier result of G. Grtzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A⊠B introduced in Part I of this paper.
lattice of completely regular semigroup varieties , J. Austral. Math. Soc. Ser. A , 49 ( 1990 ), 24 – 42 . [4] Pastijn , F. and Yan , X
Abstract
G. Grtzer and F. Wehrung introduced the lattice tensor product, A⊠B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A⊠B, that is, represent A⊠,B as a subset A<B> of B A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A<B> ≌ (Con A)<Con B>, which is a special case of a recent result of G. Grtzer and F. Wehrung and a generalization of a 1981 result of G. Grtzer, H. Lakser, and R.W. Quackenbush.
Abstract
We present a new gluing construction for semimodular lattices, related to the Hall-Dilworth construction
. , The reflexive dimension of a lattice polytope , Ann. Comb. , 10 ( 2006 ), no. 2 , 211 – 217 . [16] Ohsugi , H. and Hibi , T
, G. , Abelian groups determined by subgroup lattices of direct powers, Arch. Math. (Basel), 86 (2), (2006), 97–100. MR 2006j :20082 Călugăreanu G. Abelian groups
