Enumeration under group action

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.

Relative blocking in posets

Generalized arithmetic function algebras

A sheaf of bicomodules over the incidence coalgebra

In terminology of number theory, an arithmetic incidence function can be characterized as a function that possesses all the defining properties of both an incidence function and an arithmetic function of two variables. Under this characterization, a function f : Z+×Z+ → C is an arithmetic incidence function if f(x, y) = 0 whenever the element x does not preceed the element y, where the order is determined by the standard ordering of positive integers. Any suborder of the standard ordering of the positive integers determines a subclass of arithmetic incidence functions specific to that suborder. This thesis concentrates on the subclass which is determined by the divisibility ordering of positive integers. The primary objective is to present arithmetic incidence functions as a natural generalization of arithmetic functions of one variable, and to generalize the associated notions of multiplicativity and complete multiplicativity together with some basic results into the context of arithmetic incidence functions. The secondary objective is to present connections and differences between the theory of incidence functions as generalized arithmetic functions and the classical theory of arithmetic functions. Two types of convolutions, namely the D-convolution and the C -convolution of arithmetic incidence functions, are introduced as generalizations of the Dirichlet convolution and the unitary convolution of arithmetic functions, respectively. Also the related Mobius functions are presented. The main theme is the factorability of arithmetic incidence functions, where the scope of the term covers a set of related properties. The translation invariance and the complete translation invariance, although not to be regarded as actual concepts of factorability as such, have an important role as elements of factorability and complete factorability. The concepts of semifactorability and complete semifactorability are presented as generalizations of multiplicativity and complete multiplicativity of an arithmetic function. These two concepts are also elements of factorability and complete factorability, which for their part are presented as alternative generalizations of multiplicativity and complete multiplicativity. In addition, the dual concepts for the aforementioned factorability concepts are introduced.

Arithmetic Incidence Functions : A Study of Factorability

/pdf/multiplication-operators-on-incidence-algebras-4c8yzezmpm.pdf

Multiplication Operators on Incidence Algebras

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Multiplication Operators on Incidence Algebras