A normal form theorem for lattices completely generated by a subset

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Abstract
For an m \mathfrak {m} -complete lattice L ( m \mathfrak {m} is an infinite regular cardinal) and subset X of L that m \mathfrak {m} -generates L, we prove a Normal Form Theorem for elements of L expressed as polynomials over X. This generalizes a theorem of B. Jónsson in which such a representation is found for the lattice L freely m \mathfrak {m} -generated by a poset X. We also apply this result to free m \mathfrak {m} -products of m \mathfrak {m} -complete lattices.

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