2-valued morphism

From Wikipedia, the free encyclopedia

2-valued morphism is a term used in mathematics to describe a morphism that sends a Boolean algebra $B$ onto a Boolean algebra $2$ which consists of only two elements (i.e. one distinction): $I$ and $O$ .

They can be interpreted as representing a particular state of $B$ . All propositions of $B$ which are sent upon $I$ are assumed to be true, all propositions sent upon $O$ are assumed to be false. Since this morphism conserves the Boolean structures (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 that:

$s 2 * f = s 1$

The possible states of different objects defined by this way can be conceived as representing potential events. The set of events can then be structured in the some 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 chonological precendence relations. These relations would then determine a temporal order, a topology, and possibly a metric.

According to ^[1], "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" ^[1].

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.