In mathematics, a locally finite poset is a
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
''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. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
''. 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'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
a locally finite poset is also called a
causal set and has been used as a model for
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
.
References
*
Stanley, Richard P. Enumerative Combinatorics, Volume I. Cambridge University Press, 1997. Pages 98, 113–116.
Order theory
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