Winkler Conditions

Conditions arising in the study of the Robbins axiom and its connection with Boolean algebra. Winkler studied Boolean conditions (such as idempotence or existence of a zero) which would make a Robbins algebra become a Boolean algebra. Winkler showed that each of the conditions

  exists C, exists D,C v D=C
  exists C, exists D,!(C v D)=!C

where A v B denotes OR and !A denotes NOT, known as the first and second Winkler conditions, suffices. A computer proof demonstrated that every Robbins algebra satisfies the second Winkler condition, from which it follows immediately that all Robbins algebras are Boolean.

See also

Boolean Algebra, Huntington Axiom, Robbins Algebra, Robbins Axiom

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McCune, W. "Robbins Algebras are Boolean.", S. "Robbins Algebra: Conditions that Make a Near-Boolean Algebra Boolean." J. Automated Reasoning 6, 465-489, 1990.Winkler, S. "Absorption and Idempotency Criteria for a Problem in Near-Boolean Algebra." J. Algebra 153, 414-423, 1992.

Referenced on Wolfram|Alpha

Winkler Conditions

Cite this as:

Weisstein, Eric W. "Winkler Conditions." From MathWorld--A Wolfram Web Resource.

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