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Structure
Published in Michael H. Albert, Richard J. Nowakowski, David Wolfe, Lessons in Play, 2019
Michael H. Albert, Richard J. Nowakowski, David Wolfe
Many of the games constructed in this way will not be in canonical form, so we expect that the actual value of gn will be much less than that provided by this estimate. On day 2, the observation states that there are at most 28 = 256 games. But, note that if two comparable options are available to a player, one of the two will be dominated. Consequently, candidate sets of left or right options are antichains of games born by day n − 1. An antichain is a set consisting only of incomparable elements. There are six antichains of games born by day 1: {1},{0,∗},{0},{∗},{−1},{}.
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
An antichain is a poset in which no two elements are comparable (i.e., x≼y $ x{ \preccurlyeq }y $ if and only if x = y for all x and y in the antichain). A maximal chain is a chain that is not properly contained in another chain (and similarly for a maximal antichain).
On pattern setups and pattern multistructures
Published in International Journal of General Systems, 2020
Aimene Belfodil, Sergei O. Kuznetsov, Mehdi Kaytoue
A partial order on a set P is a binary relation ≤ on P that is reflexive (), transitive ( if and then .) and antisymmetric ( if and then x = y). The pair is called a partially ordered set or a poset for short. Two elements x and y from P are said to be comparable if or ; otherwise, they are said to be incomparable. A subset is said to be a chain (resp., an antichain) if all elements of S are pairwise comparable (resp., incomparable). The set of all chains (resp., antichains) of P is denoted by (resp., ). (Finite) posets are generally depicted by so-called (Hasse) diagrams based on the covering relation of the partial order. Figure 2(1) depicts the diagram of the powerset of a set ordered by set inclusion (i.e. poset ). In this paper, with a slight abuse of notation, we will present some infinite posets using diagrams. Figure 2(2) presents a poset where ,, and.