An analog of the fundamental theorem of arithmetic in ordered groupoids

In the note we consider ordered groupoids with the Riesz interpolation property, that is, ifa _i ≤b _j (i, j=1,2), then there exists ac such thata _i ≤c≤b _j (i, j=1,2). For such groupoids possessing the descending chain condition for the positive cone and the property

Comparison of Picard groups in dimension 1

We compare two Picard groups in dimension 1. Our proofs are constructive and the results generalize a theorem of J. Sands [11]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monoids and decay

We propose some formalization of the concept of critical decay number and describe the class of models with this number at most 4 (i.e., every object is decomposable which consists of four or more elements).     

Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).     

Factorization theory of root closed monoids of small rank

In this note, we investigate ideal and factorization-theoretic properties of some root closed cancellative commutative monoids of rank at most two.     

A Hopf algebra for counting cycles

Cycles, also known as self-avoiding polygons, elementary circuits or simple cycles, are closed walks which are not allowed to visit any vertex more than once. We present an exact formula for enumerating such cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from a Hopf algebra, which we construct explicitly, and which provides further means of counting cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate cycles.     

On graphs with a given endomorphism monoid

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M . In this paper we give a construction G(M) for a graph with prescribed endomorphism monoid M . Using this construction we derive bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. For example we show that for every monoid M , | M |=m there is a graph G with End(G)? M and |E(G)|≤(1 + 0(1))m². This is, up to a factor of 1/2, best possible since there are monoids requiring a graph with \begin{eqnarray*} && \frac{m^{2}}{2}(1 -0(1)) \end{eqnarray*} edges. We state bounds for the class of all monoids as well as for certain subclasses—groups, k‐cancellative monoids, commutative 3‐nilpotent monoids, rectangular groups and completely simple monoids. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 241–262, 2009     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .