2-valued morphism
2-valued morphism is a term used in mathematics[1] 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 s1 and s2 of B, represented by 2-valued morphisms, can then be represented by an automorphism f from B to B, such tuhat s2 o f = s1.
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 s1 to event s2. 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,[2] "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".[2]
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
- ↑ Fleischer, Isidore (1993), "A Boolean formalization of predicate calculus", Algebras and orders (Montreal, PQ, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 389, Kluwer Acad. Publ., Dordrecht, pp. 193–198, MR 1233791.
- 1 2 Heylighen, Francis (1990). A Structural Language for the Foundations of Physics. Brussels: International Journal of General Systems 18, p. 93-112.