×

zbMATH — the first resource for mathematics

Freely adjoining a complement to a lattice. (English) Zbl 1164.06308
Summary: For a bounded lattice \(K\) and an element \(a\) of \(K - \{0,1\}\), we directly describe the structure of the lattice freely generated by \(K\) and an element \(u\) subject to the requirement that \(u\) be a complement of \(a\). Earlier descriptions of this lattice used multi-step procedures.
As an application, we give a short and direct proof of the classical result of R. P. Dilworth (1945): Every lattice can be embedded into a uniquely complemented lattice. We prove it in the stronger form due to C. C. Chen and G. Grätzer [J. Algebra 11, 56–63 (1969; Zbl 0185.03701)]: Every at most uniquely complemented bounded lattice has a \(\{0,1\}\)-embedding into a uniquely complemented lattice.

MSC:
06B25 Free lattices, projective lattices, word problems
06C15 Complemented lattices, orthocomplemented lattices and posets
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] ADAMS M. E- SICHLER J.: Cover set lattices. Canad. J. Math. 32 (1980), 1177-1205. · Zbl 0423.06007
[2] ADAMS M. E.-SICHLER J.: Lattices with unique complementation. Pacific. J. Math. 92 (1981), 1-13. · Zbl 0468.06005
[3] CHEN C. C.-GRÄTZER G.: On the construction of complemented lattices. J. Algebra 11 (1969), 56-63. · Zbl 0185.03701
[4] CRAWLEY P.-DILWORTH R. P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0494.06001
[5] DEAN R. A.: Free lattices generated by partially ordered sets and preserving bounds. Canad. J. Math. 16 (1964), 136-148. · Zbl 0122.25801
[6] DILWORTH R. P.: Lattices with unique complements. Trans. Amer. Math. Soc. 57 (1945), 123-154. · Zbl 0060.06103
[7] GRÄTZER G.: General Lattice Theory. (2nd, Birkhäuser Verlag, Basel, 1998 · Zbl 0909.06002
[8] GRÄTZER G.: A reduced free product of lattices. Fund. Math. 73 (1971/72), 21-27. · Zbl 0229.06002
[9] GRÄTZER G.-LAKSER H.: Freely adjoining a relative complement to a lattice. Algebra Universalis 53 (2005), 189-210. · Zbl 1083.06011
[10] GRÄTZER G.-LAKSER H.: Embedding lattices into m-transitively complemented lattices. · Zbl 0772.08003
[11] GRÄTZER G.-LAKSER H.-PLATT C. R.: Free products of lattices. Fund. Math. 69 (1970), 233-240. · Zbl 0206.29703
[12] HUNTINGTON E. V.: Sets of independent postulates for the algebra of logic. Trans. Amer. Math. Soc. 79 (1904), 288-309. · JFM 35.0087.02
[13] LAKSER H.: Free lattices generated by partially ordered sets. Ph.D. Thesis, University of Manitoba, 1968.
[14] SALII V. N.: Lattices with Unique Complements. Transl. Math. Monogr. 69, Amer. Math. Soc, Providence, RI. · Zbl 0632.06009
[15] WHITMAN P. M.: Free lattices. I; II. Ann. of Math. (2) 42; 43 (1941; 1942), 325-330; 104-115. · Zbl 0024.24501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.