Incidence algebra

In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered setand commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. ζ 2 ( x , y )   =   ∑ z ∈ [ x , y ] ζ ( x , z ) ζ ( z , y )   =   ∑ z ∈ [ x , y ] 1   =   # [ x , y ] . {displaystyle extstyle zeta ^{2}(x,y) = sum _{zin }zeta (x,z),zeta (z,y) = sum _{zin }1 = #.} t n ( S , T )   =   ∑ t ( T 0 , T 1 ) t ( T 1 , T 2 ) ⋯ t ( T n − 1 , T n )   =   { n ! if    | T ∖ S | = n 0 otherwise, {displaystyle t^{n}(S,T) = sum t(T_{0},T_{1}),t(T_{1},T_{2})cdots t(T_{n-1},T_{n}) = left{{egin{array}{cl}n!&{ ext{if }} |T{setminus }S|=n\0&{ ext{otherwise,}}end{array}} ight.} μ   =   1 ζ   =   exp ⁡ ( − t )   =   ∑ n ≥ 0 ( − 1 ) n t n n ! . {displaystyle extstyle mu = {frac {1}{zeta }} = exp(-t) = sum _{ngeq 0}(-1)^{n}{frac {t^{n}}{n!}}.} ( δ n δ m ) ( a , b )   =   ∑ a | c | b δ n ( a , c ) δ m ( c , b )   =   δ n m ( a , b ) , {displaystyle (delta _{n}delta _{m})(a,b) = sum _{a|c|b}delta _{n}(a,c),delta _{m}(c,b) = delta _{nm}(a,b),} μ ( n ) = μ D ( 1 , n ) = ∏ k ≥ 1 μ N ( 0 , e k ) = { ( − 1 ) d  for  n  squarefree with  d  prime factors 0 otherwise. {displaystyle mu (n),=,mu _{D}(1,n),=,prod _{kgeq 1}mu _{mathbb {N} }(0,e_{k}),=,left{{egin{array}{cl}(-1)^{d}&{ ext{ for }}n{ ext{ squarefree with }}d{ ext{ prime factors}}\0&{ ext{otherwise.}}end{array}} ight.} In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered setand commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. A locally finite poset is one in which every closed interval is finite. The members of the incidence algebra are the functions f assigning to each nonempty interval a scalar f(a, b), which is taken from the ring of scalars, a commutative ring with unity. On this underlying set one defines addition and scalar multiplication pointwise, and 'multiplication' in the incidence algebra is a convolution defined by