A normal form theorem for lattices completely generated by a subset

Abstract:

For an $\mathfrak {m}$-complete lattice L ($\mathfrak {m}$ is an infinite regular cardinal) and subset X of L that $\mathfrak {m}$-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 $\mathfrak {m}$-generated by a poset X. We also apply this result to free $\mathfrak {m}$-products of $\mathfrak {m}$-complete lattices.