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A theorem on partially ordered sets of order-preserving mappings
| Content Provider | Semantic Scholar |
|---|---|
| Author | Duffus, Dwight Wille, Rudolf |
| Copyright Year | 1979 |
| Abstract | Let P be a partially ordered set and let PP denote the set of all order-preserving mappings of P to P ordered byf b in a partially ordered set X, a covers b if a > c > b implies a = c. Let V(X) denote the set of elements of X with a unique lower cover in X. Let Xd denote the dual of X. The following result is due to D. Duffus and I. Rival [3]. Logarithmic property. Let X and Y be finite partially ordered sets. If X has a least element, then V(X Y) _ V(X) X yd. Let X be a finite partially ordered set. For x E X we let [x) = {y E Xly ) x) and let V(x) = V([x)). We also let the length l(X) of X be defined by I(X) = sup{ i Cl 1 C C X, C is a chain} and the depth 6(x) of x in X is given by 6(x) = sup{ l(C)I C C X, C is a chain and inf(C) = x}. It is clear that x has maximum depth in X if and only if 6(x) = I(X). Received by the editors February 16, 1978. AMS (MOS) subject classifications (1970). Primary 06A10. 'The work presented here was supported in part by the National Research Council of Canada. ? 1979 American Mathematical Society 0002-9939/79/0000-0353/$01.75 14 This content downloaded from 40.77.167.66 on Fri, 08 Jul 2016 05:59:07 UTC All use subject to http://about.jstor.org/terms THEOREM ON PARTIALLY ORDERED SETS OF MAPPINGS 15 Let P and Q be finite, connected partially ordered sets and let p be an isomorphism of PP onto Q Q. Let f E PP satisfy 6(f) = I(P P). Since l(X Y) = I(X) I YI for finite partially ordered sets X and Y [3], 6(f) = I(P) IPI. Moreover, 8 (f IRiI IV(s)I= IRiI R31 ITi. Since t M V(t), V(t) is a proper subset of T. Therefore, R1 = 0 or R3 = 0. Let us suppose that R1 = 0. Then V(t) = 0 and, because t is a minimal element of the connected partially ordered set T, I TI = 1. Hence, from (3), uu p Pp QQ _ (SSU)(UU)s It follows that ISI = 1 and that PU Q. In the case that R3 = 0, a similar argument establishes that P Q. The proof of the theorem is now complete. We observe that the same argument as used in the proof of the Theorem verifies a curious "cancellation" law. This content downloaded from 40.77.167.66 on Fri, 08 Jul 2016 05:59:07 UTC All use subject to http://about.jstor.org/terms 16 DWIGHT DUFFUS AND RUDOLF WILLE COROLLARY. Let P and Q be finite, connected partially ordered sets. Then PQ QP implies P Q. The corollary does not hold for all finite antichains n: 24 _ 4!? |
| Starting Page | 14 |
| Ending Page | 16 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1979-0534380-2 |
| Volume Number | 76 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1979-076-01/S0002-9939-1979-0534380-2/S0002-9939-1979-0534380-2.pdf |
| Alternate Webpage(s) | https://www.ams.org/journals/proc/1979-076-01/S0002-9939-1979-0534380-2/S0002-9939-1979-0534380-2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
