In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a 2-valued morphism
[.] is a
homomorphism that sends a
Boolean algebra ''B'' onto the
two-element Boolean algebra 2 = . It is essentially the same thing as an
ultrafilter on ''B'', and, in a different way, also the same things as a
maximal ideal of ''B.'' 2-valued morphisms have also been proposed as a tool for unifying the language of physics.
2-valued morphisms, ultrafilters and maximal ideals
Suppose ''B'' is a Boolean algebra.
* If ''s'' : ''B'' → 2 is a 2-valued morphism, then the set of elements of ''B'' that are sent to 1 is an ultrafilter on ''B'', and the set of elements of ''B'' that are sent to 0 is a maximal ideal of ''B''.
* If ''U'' is an ultrafilter on ''B'', then the complement of ''U'' is a maximal ideal of ''B'', and there is exactly one 2-valued morphism ''s'' : ''B'' → 2 that sends the ultrafilter to 1 and the maximal ideal to 0.
* If ''M'' is a maximal ideal of ''B'', then the complement of ''M'' is an ultrafilter on ''B'', and there is exactly one 2-valued morphism ''s'' : ''B'' → 2 that sends the ultrafilter to 1 and the maximal ideal to 0.
Physics
If the elements of ''B'' are viewed as "propositions about some object", then a 2-valued morphism on ''B'' can be interpreted as representing a particular "state of that object", namely the one where the propositions of ''B'' which are mapped to 1 are true, and the propositions mapped to 0 are false. Since the morphism conserves the Boolean operators (
negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
,
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 true propositions form an ultrafilter, the false propositions form a maximal ideal, as mentioned above.)
The transition between two states ''s''
1 and ''s''
2 of ''B'', represented by 2-valued morphisms, can then be represented by an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
''f'' from ''B'' to ''B'', such that ''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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
, a
topology, and possibly a
metric.
According to,
"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 distinction states as events, or morphisms as processes, is much less straightforward.
References
{{reflist
External links
"Representation and Change - A metarepresentational framework for the foundations of physical and cognitive science"
Boolean algebra