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Abstract  

In 1999, for lattices A and B, G. Grtzer and F. Wehrung introduced the lattice tensor product, AB. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize AB.

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Abstract  

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Gr�tzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.

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Summary G. Grätzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Grätzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders.

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