Abstract
These lectures notes present the theory of free lattices, assuming only a basic understanding of lattice theory. They begin with Whitman’s solution to the word problem and his canonical form. The well known consequences of these are given as well as several lesser known consequences, such as the continuity of free lattices, the existence of a fixed point free unary polynomial on a free lattice, and the fact that finite sub-lattices of a free lattice satisfy a nontrivial lattice equation. The theory of covers in free lattices is developed and some of the consequences explored. Tschantz’s Theorem and a new characterization of semisingular elements are discussed and some important consequences of these results are given such as the existence of dense maximal chains in intervals of a free lattice.
The author would like to thank Jennifer Hyndman upon whose careful notes these lecture notes are based. She made numerous suggestions which greatly improved the readability of these notes. She is also responsible for many of the TEXnical details of this manuscript.
The author would like to thank J. B. Nation and J. Ježek for many helpful suggestions. He would also like to thank No Rosenberg, the Université de Montréal, and NATO for inviting him to this conference.
This research was supported by NSF Grant no. DMS89-01756.
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Freese, R. (1993). Lectures on Free Lattices. In: Rosenberg, I.G., Sabidussi, G. (eds) Algebras and Orders. NATO ASI Series, vol 389. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0697-1_5
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DOI: https://doi.org/10.1007/978-94-017-0697-1_5
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