Median algebra (1980)

A study of algebras with a ternary operation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,\,y,\,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying some identities, equivalent to embeddability in a lattice with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,\,y,\,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> realized as, simultaneously, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x logical-and left-parenthesis y logical-or z right-parenthesis right-parenthesis logical-or left-parenthesis y logical-and z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x\, \wedge \,(y\, \vee \,z))\, \vee \,(y\, \wedge \,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x logical-or left-parenthesis y logical-and z right-parenthesis right-parenthesis logical-and left-parenthesis y logical-or z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x\, \vee \,(y\, \wedge \,z))\, \wedge \,(y\, \vee \,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is weaker than embeddability in a modular lattice, where those expressions coincide for all <italic>x</italic>, <italic>y</italic>, and <italic>z</italic>, 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-<italic>cells</italic>"). In distributive lattices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,\,y,\,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduces to the median <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x logical-and y right-parenthesis logical-or left-parenthesis x logical-and z right-parenthesis logical-or left-parenthesis y logical-and z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>∨</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>∧</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x\, \wedge \,y)\, \vee \,(x\, \wedge \,z)\, \vee \,(y\, \wedge \,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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.