Cardinal representations for closures and preclosures
Authors:
F. Galvin, E. C. Milner and M. Pouzet
Journal:
Trans. Amer. Math. Soc. 328 (1991), 667693
MSC:
Primary 06A06; Secondary 04A20, 06A15, 54A05
MathSciNet review:
1016806
Fulltext PDF Free Access
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]
Garrett
Birkhoff, Lattice theory, Third edition. American Mathematical
Society Colloquium Publications, Vol. XXV, American Mathematical Society,
Providence, R.I., 1967. MR 0227053
(37 #2638)
 [2]
P.
M. Cohn, Universal algebra, 2nd ed., Mathematics and its
Applications, vol. 6, D. Reidel Publishing Co., DordrechtBoston,
Mass., 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]
Dwight
Duffus and Maurice
Pouzet, Representing ordered sets by chains, Orders:
description and roles (L’Arbresle, 1982) NorthHolland Math. Stud.,
vol. 99, NorthHolland, Amsterdam, 1984, pp. 81–98
(English, with French summary). 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 (1961/1962), 123–128. MR 0131986
(24 #A1833)
 [7]
Keith
R. Milliken, Completely separable families and Ramsey’s
theorem, J. Combinatorial Theory Ser. A 19 (1975),
no. 3, 318–334. MR 0403983
(53 #7792)
 [8]
E.
C. Milner and M.
Pouzet, On the cofinality of partially ordered sets, Ordered
sets (Banff, Alta., 1981) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci.,
vol. 83, Reidel, DordrechtBoston, Mass., 1982,
pp. 279–298. MR 661297
(83k:06003)
 [9]
E.
C. Milner and M.
Pouzet, On the independent subsets of a closure system with
singular dimension, Algebra Universalis 21 (1985),
no. 1, 25–32. MR 835967
(87f:06005), http://dx.doi.org/10.1007/BF01187553
 [10]
Z. Nagy and Z. Szentmiklóssy, On the representation of tournaments, preprint.
 [11]
Stevo
Todorčević, Directed sets and cofinal
types, Trans. Amer. Math. Soc.
290 (1985), no. 2,
711–723. MR
792822 (87a:03084), http://dx.doi.org/10.1090/S00029947198507928229
 [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), 161166. MR 0032578 (11:309f)
 [4]
 D. Duffus and M. Pouzet, Representing ordered sets by chains, Ann. Discrete Math. 23 (1984), 8198. MR 779846 (86f:06001)
 [5]
 P. Erdös and A. Tarski, On families of mutually exclusive sets, Ann. of Math. (2) 44 (1943), 315329. MR 0008249 (4:269b)
 [6]
 A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961), 123128. MR 0131986 (24:A1833)
 [7]
 Keith R. Milliken, Completely separable families and Ramsey's theorem, J. Combin. Theory Ser. A 19 (1975), 318334. 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. 279298. MR 661297 (83k:06003)
 [9]
 , On the independent subsets of a closure system with singular dimension, Algebra Universalis 21 (1985), 2532. 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), 711723. 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:
http://dx.doi.org/10.1090/S00029947199110168063
PII:
S 00029947(1991)10168063
Keywords:
Cardinal number,
directed graph,
quasiorder,
algebraic and topological closures,
lattice,
breadth,
independent set
Article copyright:
© Copyright 1991
American Mathematical Society
