Lattice Tensor Products i. Coordinatization
Tóm tắt
G. Grätzer and F. Wehrung introduced the lattice tensor product, A ⊠ B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A ⊠ B is a congruence-preserving extension of B. The lattice A ⊠ B is defined as the set of certain subsets of A ⊠ B; there is no easy test when a subset belongs to A ⊠ B. A special case, M
3⊠B, was earlier defined by G. Gräatzer and F. Wehrung as M
3, the it Boolean triple construct, defined as a subset of B
3, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A, we can ``coordinatize"" A ⊠ B, that is, represent A ⊠ B as a subset of B
n (where n is the number of join-irreducible elements of A), and provide an effective criteria to recognize the n-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we reprove a special case of the above result: for a finite simple lattice A, the lattice A ⊠ B is a congruence-preserving extension of B.
Tài liệu tham khảo
G. Grätzer and E. T. Schmidt, Regular congruence-preserving extensions, Algebra Universalis, 46 (2001), 119–130.
G. Grätzer and F. Wehrung, The M 3[D] construction and n-modularity, Algebra Universalis, 41 (1999), 87–114.
R. W. Quackenbush, Non-modular varieties of semimodular lattices with a spanning M 3, special volume on ordered sets and their applications (L'Arbresle, 1982). Discrete Math., 53 (1985), 193–205.
E. T. Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied., 18 (1968), 3–20.