κ-Complete Uniquely Complemented Lattices

Order - 2008
John Harding1
1Department of Mathematical Sciences, New Mexico State University, Las Cruces, USA

Tóm tắt

We show that for any infinite cardinal κ, every complete lattice where each element has at most one complement can be regularly embedded into a uniquely complemented κ-complete lattice. This regular embedding preserves all joins and meets, in particular it preserves the bounds of the original lattice. As a corollary, we obtain that every lattice where each element has at most one complement can be embedded into a uniquely complemented κ-complete lattice via an embedding that preserves the bounds of the original lattice.

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Tài liệu tham khảo

Adams, M.E.: Uniquely complemented lattices. In: Bogart, K., Freese, R. Kung, J. (eds.) The Dilworth Theorems: Selected Papers of Robert P. Dilworth, pp. 79–84. Birkhäuser, Boston (1990)

Adams, M.E., Sichler, J.: Cover set lattices. Can. J. Math. 32, 1177–1205 (1980)

Adams, M.E., Sichler, J.: Lattices with unique complementation. Pac. J. Math. 92, 1–13 (1981)

Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Coll. Publ. XXV, 3rd edn. American Mathematical Society, Providence (1967)

Chen, C.C., Grätzer, G.: On the construction of complemented lattices. J. Algebra 11, 56–63 (1969)

Dean, R.A.: Free lattices generated by partially ordered sets and preserving bounds. Can. J. Math. 16, 136–148 (1964)

Dilworth, R.P.: Lattices with unique complements. Trans. Am. Math. Soc. 57, 123–154 (1945)

Grätzer, G.: Two problems that shaped a century of lattice theory. Not. Am. Math. Soc. 54(6), 696–707 (2007)

Grätzer, G., Lakser, H.: Freely adjoining a relative complement to a lattice. Algebra Univers. 53(2), 189–210 (2005)

Grätzer, G., Lakser, H.: Freely adjoining a complement to a lattice, manuscript. http://www.maths.umanitoba.ca/homepages/gratzer

Harding, J.: The MacNeille completion of a uniquely complemented lattice. Can. Math. Bull. 37(2), 222–227 (1994)

Huntington, E.V.: Sets of independent postulates for the algebra of logic. Trans. Am. Math. Soc. 79, 288–309 (1904)

Kunen, K.: Set Theory. An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, 102. North-Holland, Amsterdam (1980)

Saliĭ, V.N.: Lattices with Unique Complements. Translations of the Amer. Math. Soc. American Mathematical Society, Providence (1988)

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