Lexicographic product decompositions of cyclically ordered groups

Semidirect Product Decompositions of Ordered Groupoids and Inverse Semigroups

Product decompositions of quasirandom groups and a Jordan type theorem

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with |B| > |G|/k 1 3 we have B = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much sim...

Pattern avoidance in cyclically ordered structures

We generalize the notion of pattern avoidance to arbitrary functions on ordered sets, and consider specifically three scenarios for permutations: linear, cyclic and hybrid, the first one corresponding to classical permutation avoidance. The cyclic modification allows for circular shifts in the entries. Using two bijections, both ascribable to both Deutsch and Krattenthaler independently, we sin...

Minmax relations for cyclically ordered digraphs

We prove a range of minmax theorems about cycle packing and covering in digraphs whose vertices are cyclically ordered, a notion promoted by Bessy and Thomassé in their beautiful proof of the following conjecture of Gallai: the vertices of a strongly connected digraph can be covered by at most as many cycles as the stability number. The results presented here provide relations between cycle pac...

Formal power series with cyclically ordered exponents

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in ...