Median algebra

Author:
John R. Isbell

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

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Abstract: A study of algebras with a ternary operation satisfying some identities, equivalent to embeddability in a lattice with realized as, simultaneously, and . 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 reduces to the median , 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0574784-8

Keywords:
Median,
lattice,
finite geometry

Article copyright:
© Copyright 1980
American Mathematical Society