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PERFECT CODES ON SOME ORDERED SETS
Lee, Jeh-Gwon Korean Mathematical Society 2006 대한수학회보 Vol.43 No.2
Using the concept of codes on ordered sets introduced by Brualdi, Graves and Lawrence, we consider perfect codes on the ordinal sum of two ordered sets, the standard ordered sets and the disjoint sum of two chains.
ON THE DIMENSION OF AMALGAMATED ORDERED SETS
Lee, Jeh Gwon,Kim, Myung-Hwan 대한수학회 1992 대한수학회보 Vol.29 No.1
The dimension problem has been one of central themes in the theory of ordered sets. In this paper we focus on amalgamated ordered sets. Although some results can be obviously applied to infinite cases, we assume throughout that all ordered set are finite. If A and B are ordered sets whoes orders agree on A∩B, then the amalgam of A and B is defined to be the set A∪B in which the order is the transitive closure of the union of the two orders, i.e., the smallest order containing the two orders, and is denoted by A∨B. We then may have a naive conjecture that dim A∨B≤dim A+dim B for any ordered sets A and B. But it is quite surprising that the dimension of the amalgam of certain 2-dimensional ordered sets can be arbitrarily large. In fact, we have two interseting examples. EXAMPLES. 1) Let S_n be the n-dimensional standard ordered set and U_n the ordered set obtained from S_n by subdividing every edge. Although U_n is of dimension n (see Lee, Liu, Nowakowski and Rival [3]), it is the amalgam of two 2-dimensional subsets A and B, where A consists of the maximal elements and the new ones and B consists of the minimal elements and the new ones. This was pointed out by H. A. Kierstead. 2) Let us consider the following particular subset T_n of the lattice of all subsets of X = {1,2,…,n} which can be found in Lee, Liu, Nowakowski and Rival [3]: T_n = {{1},{2},…,{n},{1,2},{2,3},…,{n,1},…,X-{n},X-{1},…,X-{n-1}}. Clearly, all minimal elements and all maximal elements of T_n sonstitute the n-dimensional standard ordered set so that T_n is of dimension n. Now we divide T_n into two 2-dimensional subsets A and B whose amalgam is T_n itself : A = {{1},{2},…,{n},{1,2},{2,3},…,{n-1,n},…,X-{n},X-{1}}, B = {{n},{1},{n-1,n},{n,1},{1,2},…,X-{1},X-{2},…,X-{n}}. The following figure for n = 5 illustrates this example, where dots correspond to A∩B.
Bae, Deok Rak,Lee, Jeh Gwon 대한수학회 1994 대한수학회지 Vol.31 No.4
A lattice is called bounded if it has both the least element and the largest element which are usually denoted by 0 (zero) and 1 (unit), respectively. Recently, Bennett [2] defined the rectangular product of two bounded lattices L_1 and L_2 to be the set {(x,y)∈L_1×L_2|x≠0,y≠0}∪{(0,0)} with the order induced from the direct product L_1×L_2.
A CHARACTERIZATION THEOREM FOR FAMILIES
Rhee, Min Surp,Lee, Jeh Gwon 대한수학회 1995 대한수학회지 Vol.32 No.1
Let P be a finite ordered set and a∈P. Then we define the following numerical functions (Rhee [1],[2]) : f₁(a) = |{x ∈ P|x >a}|, the number of descendants of a, f₂(a) = |{x ∈ P|x < a}|, the number of ancestors of a. We call P a family if both f₁(a) > f₁(b) and f₂(a)< f₂(b) imply a<b for any elements a and b in P. In this paper we prove the following theorem which characterizes the class of all families. The notations and terminlogy in the theorem will be defined below. We assume throughout that all ordered sets are finite.
ON THE JUMP NUMBER OF SPLITS OF ORDERED SETS
Jung, Hyung-Chan,Lee, Jeh-Gwon Korean Mathematical Society 2000 대한수학회보 Vol.37 No.4
In this paper, we consider the jump number of the split P[S] of a subset S ordered set P. $For\ x\in\ P,\ we\ show\ that\ s(P)\leq\ s(P[x]\leq\ s(P)+2$ and give a necessary and sufficient condition for which s(P[x])=s(P).
WEAK DIMENSION AND CHAIN-WEAK DIMENSION OF ORDERED SETS
KIM, JONG-YOUL,LEE, JEH-GWON Korean Mathematical Society 2005 대한수학회보 Vol.42 No.2
In this paper, we define the weak dimension and the chain-weak dimension of an ordered set by using weak orders and chain-weak orders, respectively, as realizers. First, we prove that if P is not a weak order, then the weak dimension of P is the same as the dimension of P. Next, we determine the chain-weak dimension of the product of k-element chains. Finally, we prove some properties of chain-weak dimension which hold for dimension.