Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cardinal representations for closures and preclosures


Authors: F. Galvin, E. C. Milner and M. Pouzet
Journal: Trans. Amer. Math. Soc. 328 (1991), 667-693
MSC: Primary 06A06; Secondary 04A20, 06A15, 54A05
DOI: https://doi.org/10.1090/S0002-9947-1991-1016806-3
MathSciNet review: 1016806
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A cardinal representation of a preclosure $ \varphi $ on a set $ E$ is a family $ \mathcal{A} \subseteq \mathcal{P}(E)$ such that for any set $ X \subseteq \cup \mathcal{A},\varphi (X) = E$ holds if and only if $ \vert X \cap A\vert= \vert A\vert$ for every $ A \in \mathcal{A}$. We show, for example (Theorem 2.3) that any topological closure has such a representation, but there are closures which have no cardinal representation (Theorem 11.2). We prove that, if $ k$ is finite and a closure has no independent set of size $ k + 1$, then it has a cardinal representation, $ \mathcal{A}$, of size $ \vert\mathcal{A}\vert \leq k$ (Theorem 2.4). This result is used to give a new proof of a theorem of D. Duffus and M. Pouzet [4] about gaps in a lattice of finite breadth. We do not know if a closure which has no infinite independent set necessarily has a cardinal representation, but we do prove this is so for the special case of a closure on a countable set (Theorem 2.5). Even in this special case, nothing can be said about the size of the cardinal representation; however, if the closure is algebraic, then there is a finite cardinal representation (Theorem 2.6). These results do not hold for preclosures in general, but if a preclosure on a countable set has no independent set of size $ k + 1$ ($ k$ finite), then it has a cardinal representation $ \mathcal{A}$ of size $ \vert\mathcal{A}\vert \leq k$ (Theorem 2.7).


References [Enhancements On Off] (What's this?)

  • [1] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, R.I., 1967. MR 0227053 (37:2638)
  • [2] P. M. Cohn, Universal algebra, Reidel, Dordrecht, 1981. MR 620952 (82j:08001)
  • [3] R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161-166. MR 0032578 (11:309f)
  • [4] D. Duffus and M. Pouzet, Representing ordered sets by chains, Ann. Discrete Math. 23 (1984), 81-98. MR 779846 (86f:06001)
  • [5] P. Erdös and A. Tarski, On families of mutually exclusive sets, Ann. of Math. (2) 44 (1943), 315-329. MR 0008249 (4:269b)
  • [6] A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961), 123-128. MR 0131986 (24:A1833)
  • [7] Keith R. Milliken, Completely separable families and Ramsey's theorem, J. Combin. Theory Ser. A 19 (1975), 318-334. MR 0403983 (53:7792)
  • [8] E. C. Milner and M. Pouzet, On the cofinality of partially ordered sets, Ordered Sets (Banff, 1981) (I. Rival, ed.), Reidel, 1982, pp. 279-298. MR 661297 (83k:06003)
  • [9] -, On the independent subsets of a closure system with singular dimension, Algebra Universalis 21 (1985), 25-32. MR 835967 (87f:06005)
  • [10] Z. Nagy and Z. Szentmiklóssy, On the representation of tournaments, preprint.
  • [11] S. Todorčević, Directed sets and cofinal types, Trans. Amer. Math. Soc. 290 (1985), 711-723. MR 792822 (87a:03084)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06A06, 04A20, 06A15, 54A05

Retrieve articles in all journals with MSC: 06A06, 04A20, 06A15, 54A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1016806-3
Keywords: Cardinal number, directed graph, quasi-order, algebraic and topological closures, lattice, breadth, independent set
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society