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"]


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Proper morphism — In algebraic geometry, a proper morphism between schemes is an analogue of a proper map between topological spaces. Contents 1 Definition 2 Examples 3 Properties and characterizations of proper morphisms …   Wikipedia

  • Vector-valued differential form — In mathematics, a vector valued differential form on a manifold M is a differential form on M with values in a vector space V . More generally, it is a differential form with values in some vector bundle E over M . Ordinary differential forms can …   Wikipedia

  • List of mathematics articles (0-9) — NOTOC 0 −0 (number) −1 (number) −40 (number) Σ compact space Ω consistent theory Γ convergence Δ hyperbolic space Ω logic Ε net Ε quadratic form Μ recursive function yllion 0 (number) Ε₀ 0,1 simple lattice 0.999... ( 2, 3, 7) pretzel knot (2,3,7) …   Wikipedia

  • International Journal of General Systems — Infobox Journal abbreviation = Int J Gen Syst discipline = Systems sciences editor = George J. Klir publisher = Taylor and Francis Ltd. country = United States frequency = 6 Issues per year history = 1974 present website =… …   Wikipedia

  • Sheaf (mathematics) — This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… …   Wikipedia

  • Glossary of scheme theory — This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of… …   Wikipedia

  • Heyting algebra — In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras, named after Arend Heyting. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded… …   Wikipedia

  • Functor — For functors as a synonym of function objects in computer programming to pass function pointers along with its state, see function object. For the use of the functor morphism presented here in functional programming see also the fmap function of… …   Wikipedia

  • Ringed space — In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are functions on each open set of the space. Ringed spaces appear throughout analysis and are also used to… …   Wikipedia

  • Vector bundle — The Möbius strip is a line bundle over the 1 sphere S1. Locally around every point in S1, it looks like U × R, but the total bundle is different from S1 × R (which is a cylinder instead). In mathematics, a vector bundle is a… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”