Transactions of the American Mathematical Society

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



Median algebra

Author: John R. Isbell
Journal: Trans. Amer. Math. Soc. 260 (1980), 319-362
MSC: Primary 06B05
MathSciNet review: 574784
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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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Keywords: Median, lattice, finite geometry
Article copyright: © Copyright 1980 American Mathematical Society