Intrinsic order graph

The intrinsic order, denoted by < , is a partial order relation on the set 0,1 " of all binary n-tuples. The usual representation of this kind of binary relations is the Hasse diagram (9). In particular, the Hasse diagram of the partially ordered set ( 0,1 ", <) is referred to as the intrinsic order graph for n variables. 

The Hasse diagram of the poset will be also called the intrinsic order graph for n variables, denoted as well by I . 

For small values of n, the intrinsic order graph I can be directly constructed by using either Theorem 2.2 (matrix description of the intrinsic order) or Theorem 2.12 (matrix description of the covering relation for the intrinsic order). For instance, for = /] = ( 0,1, -<), and its Hasse diagram is shown in Fig. 2.1. 

Figure 2.2 illustrates the above iterative process for the first few values of n, denoting all the binary n-tuples by their decimal equivalents. Basically, we first add to / ] its isomorphic copy 2 + 7 i. This addition must be performed by placing the powers of two, 2" and 2" at consecutive levels in the intrinsic order graph. The reason is simply that... 

For further theoretical properties and practical applications of the intrinsic order and the intrinsic order graph, we refer the reader to, e.g., (2 4-8). 

The following proposition states a duality property of the intrinsic order, which explains the symmetric structure of the intrinsic order graph. 

Next corollary provides us with two easy criteria for rapidly identifying pairs of complementary binary strings in the intrinsic order graph. 

The analysis of CSBSs can be performed by using the intrinsic ordering between binary n-tuples of Os and Is. The duality property of the intrinsic order relation for complementary n-tuples (obtained by changing Os into Is and Is into Os) implies many different properties of CSBSs. Some of these properties have been rigorously proved and illustrated by the intrinsic order graph. 

Gonzalez L (2006) A picture for complex stochastic Boolean systems the intrinsic order graph. Lect Notes Comput Sci 3993 305-312... 

Gonzalez L (2007) Algorithm comparing binary string probabilities in complex stochastic Boolean systems using intrinsic order graph. Adv Complex Syst 10(Suppl.l) lll-143... 

Gonzalez L (2011) Complex stochastic Boolean systems new proptnties of the intrinsic order graph. Lecture notes in engineering tmd computer science proceedings of the world congress on engineering 2011, WCE 2011, London, pp 1194-1199, 6-8 July 2011... 

Blocked Space-Time Patterns The intrinsic phase structure of a rule can sometimes be directly observed by graphing its blocked pattern. We can describe the space-time sets of the two ordered states of rule R18 by... 

Theorem. Any nonplanar graph which has no automorphisms of order two is intrinsically chiral [28J. 

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