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  • 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. (wikipedia.org)
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  • The multiplicative identity element of the incidence algebra is the delta function, defined by δ ( a , b ) = { 1 if a = b , 0 if a ≠ b . {\displaystyle \delta (a,b)={\begin{cases}1&{\text{if }}a=b,\\0&{\text{if }}a\neq b.\end{cases}}} The zeta function of an incidence algebra is the constant function ζ(a, b) = 1 for every nonempty interval [a, b]. (wikipedia.org)
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  • they are sometimes called unital associative algebras for clarification. (wikipedia.org)
  • In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. (wikipedia.org)
  • In mathematics , coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows ) to unital associative algebras . (magnesiumhur905.cfd)
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  • 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. (wikipedia.org)
  • A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring . (wikipedia.org)
  • Every ring is an associative algebra over its center and over the integers. (wikipedia.org)
  • Every ring is an associative Z -algebra, where Z denotes the ring of the integers . (wikipedia.org)
  • The definition is equivalent to saying that a unital associative R -algebra is a monoid object in R -Mod (the monoidal category of R -modules). (wikipedia.org)
  • thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules . (wikipedia.org)
  • The members of the incidence algebra are the functions f assigning to each nonempty interval [a, b] a scalar f(a, b), which is taken from the ring of scalars, a commutative ring with unity. (wikipedia.org)
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  • On this underlying set one defines addition and scalar multiplication pointwise, and "multiplication" in the incidence algebra is a convolution defined by ( f ∗ g ) ( a , b ) = ∑ a ≤ x ≤ b f ( a , x ) g ( x , b ) . {\displaystyle (f*g)(a,b)=\sum _{a\leq x\leq b}f(a,x)g(x,b). (wikipedia.org)
  • The incidence algebra of ≤ is then isomorphic to the algebra of upper-triangular matrices with this prescribed zero-pattern and arbitrary (including possibly zero) scalar entries everywhere else, with the operations being ordinary matrix addition, scaling and multiplication. (wikipedia.org)
  • The convolution of functions in the incidence algebra corresponds to multiplication of formal power series: see the discussion of reduced incidence algebras below. (wikipedia.org)
  • The multiplicative identity element of the incidence algebra is the delta function, defined by δ ( a , b ) = { 1 if a = b , 0 if a ≠ b . {\displaystyle \delta (a,b)={\begin{cases}1&{\text{if }}a=b,\\0&{\text{if }}a\neq b.\end{cases}}} The zeta function of an incidence algebra is the constant function ζ(a, b) = 1 for every nonempty interval [a, b]. (wikipedia.org)
  • This definition implies that the algebra, being a ring, is unital , since rings are supposed to have a multiplicative identity . (wikipedia.org)
  • The Pappus theorem has a skewer analog in which both points and lines are replaced by lines in 3-space and the incidence between a line and a point translates as the intersection of the two respective lines at right angle. (stonybrook.edu)
  • Coalgebras occur naturally in a number of contexts (for example, representation theory , universal enveloping algebras and group schemes ). (magnesiumhur905.cfd)
  • It has been suggested that Algebra homomorphism be merged into this article. (wikipedia.org)
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  • Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. (wikipedia.org)
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