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• 1
Electronic Resource
Springer
Order 8 (1991), S. 243-246
ISSN: 1572-9273
Keywords: 06A06 ; Poset ; circle order ; dimension
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The result stated in the title is proved in this note. Actually we show that S x N is not a circle order, where S={(1, 1), (1, 2), (1, 3), (2, 1), (2, 3)}. Furthermore this non-circle order is critical in the sense that (S-{x}) x N is a circle order for any x in S.
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• 2
Electronic Resource
Springer
Order 8 (1991), S. 247-265
ISSN: 1572-9273
Keywords: 05A15 ; 05A19 ; 06A06 ; 54-04 ; Quasiordered set ; partially ordered set ; topology ; connected ; separation axiom ; antichain ; generating function
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A refinement of an algorithm developed by Culberson and Rawlins yields the numbers of all partially ordered sets (posets) with n points and k antichains for n≤11 and all relevant integers k. Using these numbers in connection with certain formulae derived earlier by the first author, one can now compute the numbers of all quasiordered sets, posets, connected posets etc. with n points for n≤14. Using the well-known one-to-one correspondence between finite quasiordered sets and finite topological spaces, one obtains the numbers of finite topological spaces with n points and k open sets for n≤11 and all k, and then the numbers of all topologies on n≤14 points satisfying various degrees of separation and connectedness properties, respectively. The number of (connected) topologies on 14 points exceeds 1023.
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• 3
Electronic Resource
Springer
Order 8 (1991), S. 283-290
ISSN: 1572-9273
Keywords: 06A15 ; 68RXX ; Closure operator
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We present a unifying theoretical and algorithmic approach to the problems to determine all closed sets of a closure operator, to do this up to isomorphism, and to determine the elements of certain ideals of a power set. This will be done by generalizing the concept of closure operators using the interplay of several orders of a power set.
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• 4
Electronic Resource
Springer
Order 9 (1992), S. 127-137
ISSN: 1572-9273
Keywords: Primary 06A07 ; secondary 05A15 ; Fence ; strictly increasing mapping ; order-preserving mapping ; partition
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract LetY be a fence of sizem andr=⌊m−1/2⌊. The numberb(m) of order-preserving selfmappings ofY is equal toA r-Br-Cr-Dr, where, ifm is odd, $$\begin{gathered} A_r = 2(r + 1)\sum\limits_{s = 0}^r {\left( {\begin{array}{*{20}c} {r + s} \\ {2s} \\ \end{array} } \right)} 4^s , B_r = 2r\sum\limits_{s = 1}^r {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ {s - 1} \\ \end{array} } \right),} \hfill \\ C_r = 4r\sum\limits_{s = 0}^{r - 1} {\left( {\begin{array}{*{20}c} {r + s} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right), D_r = \sum\limits_{s = 0}^{r - 1} {(2s + 1)} \left( {\begin{array}{*{20}c} {r + s - 1} \\ s \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r - 1} \\ s \\ \end{array} } \right)} \hfill \\ \end{gathered}$$ . Ifm is even, a similar formula forb(m) is true. The key trick in the proof is a one-to-one correspondence between order-preserving selfmappings ofY and pairs consisted of a partition ofY and a strictly increasing mapping of a subfence ofY toY.
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• 5
Electronic Resource
Springer
Order 9 (1992), S. 201-201
ISSN: 1572-9273
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
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• 6
Electronic Resource
Springer
Order 9 (1992), S. 203-204
ISSN: 1572-9273
Keywords: Primary 06-04 ; secondary 06A07 ; 05C30 ; Unlabeled ; partial orders ; computer ; enumeration
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The number of non-isomorphic posets on 13 elements is P13=33,823,827,452 An announcement of this result was sent to the Abstracts of the A.M.S. on July 4th, 1992. . This extends our previous result P12 which constituted the greatest known value. A table enumerates the posets according to their number of relations.
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• 7
Electronic Resource
Springer
Order 9 (1992), S. 239-244
ISSN: 1572-9273
Keywords: 06A06 ; 08A40 ; Ternary discriminator ; dual discriminator ; near-projection ; bounded partial order ; functionally complete ; Rosenberg's completeness theorem
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Algebras with an operator that discriminates order are functionally complete.
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• 8
Electronic Resource
Springer
Order 1 (1984), S. 3-5
ISSN: 1572-9273
Keywords: Primary 06A10 ; secondary 68C25 ; Jump number ; jump-critical ordered sets
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The maximum size of a jump-critical ordered set with jump-number m is at most (m+1)!
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• 9
Electronic Resource
Springer
Order 1 (1984), S. 21-28
ISSN: 1572-9273
Keywords: 06A10 ; 05A05 ; Partially ordered sets ; Sperner's Theorem ; LYM property ; product of chains
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Let P be the poset k 1 × ... × k n , which is a product of chains, where n≥1 and k 1≥ ... ≥k n ≥2. Let $$M = k_1 - \sum\nolimits_{i = 2}^n {(k_i - 1)}$$ . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M≤1. This is a generalization of a classical result, Sperner's Theorem, which is the case k 1= ... =k n =2. We also determine the number and location of the maximum ranks of P.
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• 10
Electronic Resource
Springer
Order 1 (1984), S. 29-34
ISSN: 1572-9273
Keywords: Primary 06A10 ; secondary 05C60 ; 20B25 ; Poset ; Peck poset ; Sperner property ; automorphism group ; quotient poset ; edge-reconstruction
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract An elementary, self-contained proof of a result of Pouzet and Rosenberg and of Harper is given. This result states that the quotient of certain posets (called unitary Peck) by a finite group of automorphisms retains some nice properties, including the Sperner property. Examples of unitary Peck posets are given, and the techniques developed here are used to prove a result of Lovász on the edge-reconstruction conjecture.
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