In mathematics, a locally finite poset is a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
''P'' such that for all ''x'', ''y'' ∈ ''P'', the
interval 'x'', ''y''consists of
finitely many elements.
Given a locally finite poset ''P'' we can define its ''
incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constr ...
''. Elements of the incidence algebra are functions ''ƒ'' that assign to each interval
'x'', ''y''of ''P'' a real number ''ƒ''(''x'', ''y''). These functions form an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
with a product defined by
:
There is also a definition of ''
incidence coalgebra''.
In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
a locally finite poset is also called a
causal set and has been used as a model for
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
.
References
Stanley, Richard P. Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.
Order theory
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