A Theorem on Partially Ordered Sets of Order-preserving Mappings

Let P be a partially ordered set and let Pp denote the set of all order-preserving mappings of P to P ordered by/ < g in Pp iif(p) < g(p) for all p £ P. We prove that if P and Q are finite, connected partially ordered sets and Pp = Q<¡ then Psg. Is a partially ordered set determined by its order-preserving mappings? L. M. Gluskin [4] has shown that the set of order-preserving mappings of a partially ordered set P as a semigroup of transformations determines P up to duality. Let us consider the set PF of order-preserving mappings of P as a partially ordered set: f < g in Pp if f(p) < g(p) for all p E P. (If P is a finite lattice then P is determined by PF [6].) The purpose of this note is to establish the following. Theorem. Let P and Q be finite, connected partially ordered sets. Then Pp s*Q<2 implies P = Q. In proving the theorem, we first change the problem from one concerning exponents X Y of partially ordered sets to one about products XY (or X X Y) of partially ordered sets. The change is accomplished with a description of the covering relation in an exponent. Recall that for elements a > 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(XY) s V(X) X Yd. Let AT be a finite partially ordered set. For x E X we let [x) = {y E X\y > x) and let V(x) = V([x)). We also let the length l(X) of X be defined by l(X) = sup{|C|1|C C X, C is a chain} and the depth 8(x) of x in X is given by 8(x) = sup{/(C)|C Ç X, C is a chain and inf(C) = x). It is clear that x has maximum depth in X if and only if 8(x) = l(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 0002-9939/79/0000-035 3/$01.75 14 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THEOREM ON PARTIALLY ORDERED SETS OF MAPPINGS 15 Let P and Q be finite, connected partially ordered sets and let |Ä,||K(i)|-|Jl1||JlJ||71. Since t E V(t), V(t) is a proper subset of T. Therefore, Rx = 0 or R3 = 0. Let us suppose that Rx = 0. Then V(t) = 0 and, because / is a minimal element of the connected partially ordered set T, \T\ = 1. Hence, from (3), Uu = PP = QQ^(SSU)(UU)S. It follows that |5| = 1 and that P m U = 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. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 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 = 42! References1. G. Birkhoff,Extendedarithmetic,Duke Math. J. 3 (1937),311-316.2. D. Duffus, Toward a theory of finite partially ordered sets, Ph.D. Thesis, University ofCalgary, 1978.3. D. Duffus and I. Rival, A logarithmic property for exponents of partially ordered sets, Canad.J. Math. 30 (1978),797-807.4. L. M. Gluskin, Semigroups of isotone transformations, Uspehi Mat. Nauk 16 (1961), 157-162.MR 24 #A1336.5. J. Hashimoto, On direct product decomposition of partially ordered sets, Ann. of Math. (2) 54(1951),315-318. MR 13, 201.6. R. Wille, Cancellation and refinement results for function lattices, Houston J. Math, (toappear). Department of Mathematics, University of Calgary, Calgary, CanadaDepartment of Mathematics, Technische Hochschule Darmstadt, Darmstadt, FederalRepublic of Germany License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use