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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Median algebra
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by John R. Isbell PDF
Trans. Amer. Math. Soc. 260 (1980), 319-362 Request permission

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.
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Additional Information
  • © 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