Vol. 83, No. 1, 1979

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Chain conditions in free products of lattices with infinitary operations

George Grätzer, Andras Hajnal and David C. Kelly

Vol. 83 (1979), No. 1, 107–115

There are many facts known about the size of subsets of certain kinds in free lattices and free products of lattices. Examples: every chain in a free lattice is at most countable; every “large” subset contains an independent set; if the free product of a set of lattices contains a “long” chain, so does the free product of a flnite subset of this set of lattices. Here we investigate these problems in the setting of a variety V of m-lattices, where m is an infinite regular cardinal. An m-lattice L is a lattice in which for any nonempty set S with |S| < m, the meet and join exist in L. We obtain generalizations of many finitary results to the m-complete case. Our basic set-theoretic tool is the Erdös-Rado theorem.

Mathematical Subject Classification 2000
Primary: 06B25
Received: 21 March 1978
Revised: 29 December 1978
Published: 1 July 1979
George Grätzer
Andras Hajnal
David C. Kelly