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

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Abstract | References | Similar Articles | Additional Information

Abstract: A cardinal representation of a preclosure on a set is a family such that for any set holds if and only if for every . 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 is finite and a closure has no independent set of size , then it has a cardinal representation, , of size (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 ( finite), then it has a cardinal representation of size (Theorem 2.7).

**[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)**

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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