Tensor Products and Transferability of Semilattices

Grätzer, G. ; Wehrung, F.

Canadian journal of mathematics, Tome 51 (1999) no. 4, pp. 792-815

Résumé

In general, the tensor product, $A\otimes B$ , of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a capped tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for every lattice $B$ with zero; we shall call such lattices amenable.The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition $\left( \text{T} \right)$ .We prove that a finite lattice $A$ is amenable iff it is sharply transferable as a join-semilattice.For a general lattice $A$ with zero, we obtain the result: $A$ is amenable iff $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice.This yields, for example, that a finite lattice $A$ is amenable iff $A\otimes \text{F}\left( 3 \right)$ is a lattice iff $A$ satisfies $\left( \text{T} \right)$ , with respect to join. In particular, ${{M}_{3}}\,\otimes \,\text{F}\left( 3 \right)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

Détail

DOI : 10.4153/CJM-1999-034-6

Mots-clés : 06B05, 06B15, Tensor product, semilattice, lattice, transferability, minimal pair, capped

Bibliographie

[1] [1] Anderson, J. and Kimura, N., The tensor product of semilattices. Semigroup Forum 16(1978), 83–88. Google Scholar

[2] [2] Day, A., Herrmann, C. and Wille, R., On modular lattices with four generators. Algebra Universalis 2(1972), 317–323. Google Scholar

[3] [3] Fraser, G., The tensor product of semilattices. Algebra Universalis 8(1978), 1–3. Google Scholar

[4] [4] Freese, R., Ježek, J. and Nation, J. B., Free Lattices. Math. Surveys and Monographs 42, American Mathematical Society, Providence, Rhode Island, 1995. Google Scholar

[5] [5] Gaskill, H. S., On transferable semilattices. Algebra Universalis 2(1973), 303–316. Google Scholar

[6] [6] Gaskill, H. S., Grätzer, G. and Platt, C. R., Sharply transferable lattices. Canad. J. Math. 28(1975), 1246–1262. Google Scholar

[7] [7] Grätzer, G., Universal Algebra. 1970 Trends in Lattice Theory Sympos., U. S. Naval Academy, Annapolis,Md., Van Nostrand Reinhold, New York, 1966, 173–210. Google Scholar

[8] [8] Grätzer, G., General Lattice Theory. Second Edition. Birkhäuser Verlag, Basel. 1998. Google Scholar

[9] [9] Grätzer, G., Lakser, H. and Quackenbush, R. W., The structure of tensor products of semilattices with zero. Trans. Amer.Math. Soc. 267(1981), 503–515. Google Scholar

[10] [10] Grätzer, G. and Wehrung, F., Tensor products of semilattices with zero, revisited. J. Pure Appl. Algebra (to appear). Google Scholar

[11] [11] Grätzer, G. and Wehrung, F., The M[D] construction and n-modularity. Algebra Universalis (to appear). Google Scholar

[12] [12] Grätzer, G. and Wehrung, F., Flat semilattices. Colloq. Math. (to appear). Google Scholar

[13] [13] Jónsson, B. and Nation, J. B., A report on sublattices of a free lattice. Contributions to universal algebra, Colloq., József Attila Univ., Szeged, 1975, 223–257, Colloq. Math. Soc. János Bolyai 17, North-Holland, Amsterdam, 1977. Google Scholar

[14] [14] McKenzie, R. N., Equational bases and nonmodular lattice varieties. Trans. Amer. Math. Soc. 174(1972), 1–43. Google Scholar

[15] [15] Quackenbush, R. W., 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. Google Scholar

[16] [16] Whitman, P. M., Free lattices. Ann. of Math. (2) 42(1941), 325–330. Google Scholar