2-valued morphism

2-valued morphism

2-valued morphism is a term used in mathematicsFact|date=January 2008 to describe a morphism that sends a Boolean algebra "B" onto a two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on "B".

A 2-valued morphism can be interpreted as representing a particular state of "B". All propositions of "B" which are mapped to 1 are considered true, all propositions mapped to 0 are considered false. Since this morphism conserves the Boolean operators (negation, conjunction, etc.), the set of true propositions will not be inconsistent but will correspond to a particular maximal conjunction of propositions, denoting the (atomic) state.

The transition between two states "s"1 and "s"2 of "B", represented by 2-valued morphisms, can then be represented by an automorphism "f" from "B" to "B", such tuhat "s"2 o "f" = "s"1.

The possible states of different objects defined in this way can be conceived as representing potential events. The set of events can then be structured in the same way as invariance of causal structure, or local-to-global causal connections or even formal properties of global causal connections.

The morphisms between (non-trivial) objects could be viewed as representing causal connections leading from one event to another one. For example, the morphism "f" above leads form event "s"1 to event "s"2. The sequences or "paths" of morphisms for which there is no inverse morphism, could then be interpreted as defining horismotic or chronological precedence relations. These relations would then determine a temporal order, a topology, and possibly a metric.

According to,cite book
last = Heylighen
first = Francis
year = 1990
title = A Structural Language for the Foundations of Physics
publisher = International Journal of General Systems 18, p. 93-112
location = Brussels] "A minimal realization of such a relationally determined space-time structure can be found". In this model there are, however, no explicit distinctions. This is equivalent to a model where each object is characterized by only one distinction: (presence, absence) or (existence, non-existence) of an event. In this manner, "the 'arrows' or the 'structural language' can then be interpreted as morphisms which conserve this unique distinction".

If more than one distinction is considered, however, the model becomes much more complex, and the interpretation of distinctional states as events, or morphisms as processes, is much less straightforward.

References

External links

* [http://pcp.vub.ac.be/books/Rep&Change.pdf "Representation and Change - A metarepresentational framework for the foundations of physical and cognitive science"]


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