Abstract:A study of algebras with a ternary operation $(x, y, z)$ satisfying some identities, equivalent to embeddability in a lattice with $(x, y, z)$ realized as, simultaneously, $(x \wedge (y \vee z)) \vee (y \wedge z)$ and $(x \vee (y \wedge z)) \wedge (y \vee z)$. This is weaker than embeddability in a modular lattice, where those expressions coincide for all x, y, and z, but much of the theory survives the extension. For actual embedding in a modular lattice, some necessary conditions are found, and the investigation is carried much further in a special, geometrically described class of examples ("2-cells"). In distributive lattices $(x, y, z)$ reduces to the median $(x \wedge y) \vee (x \wedge z) \vee (y \wedge z)$, previously studied by G. Birkhoff and S. Kiss. It is shown that Birkhoff and Kiss found a basis for the laws; indeed, their algebras are embeddable in distributive lattices, i.e. in powers of the 2-element lattice. Their theory is much further developed and is connected into an explicit Pontrjagin-type duality.
- Reinhold Baer, Linear algebra and projective geometry, Academic Press, Inc., New York, N.Y., 1952. MR 0052795
- Garrett Birkhoff, Lattice Theory, American Mathematical Society, New York, 1940. MR 0001959
- Garrett Birkhoff and S. A. Kiss, A ternary operation in distributive lattices, Bull. Amer. Math. Soc. 53 (1947), 749–752. MR 21540, DOI 10.1090/S0002-9904-1947-08864-9
- Ju. L. Eršov, I. A. Lavrov, A. D. Taĭmanov, and M. A. Taĭclin, Elementary theories, Uspehi Mat. Nauk 20 (1965), no. 4 (124), 37–108 (Russian). MR 0186553
- Ju. L. Eršov and M. A. Taĭclin, Undecidability of certain theories, Algebra i Logika Sem. 2 (1963), no. 5, 37–41 (Russian). MR 0172799
- Andrzej Grzegorczyk, Undecidability of some topological theories, Fund. Math. 38 (1951), 137–152. MR 47583, DOI 10.4064/fm-38-1-137-152
- Marshall Hall, Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277. MR 8892, DOI 10.1090/S0002-9947-1943-0008892-4 J. Isbell, Some concrete dualities, Notices Amer. Math. Soc. 21 (1974), A567-A568.
- Ernest G. Manes, Algebraic theories, Graduate Texts in Mathematics, No. 26, Springer-Verlag, New York-Heidelberg, 1976. MR 0419557 H. Zassenhaus, Zum Satz von Jordan-Hölder-Schreier, Abh. Math. Sem. Univ. Hamburg 10 (1934), 106-108.
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 260 (1980), 319-362
- MSC: Primary 06B05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0574784-8
- MathSciNet review: 574784