Diameters of commuting graphs of matrices over semirings

We calculate the diameters of commuting graphs of matrices over the binary Boolean semiring, the tropical semiring and an arbitrary nonentire commutative semiring. We also find the lower bound for the diameter of the commuting graph of the semiring of matrices over an arbitrary commutative entire antinegative semiring.Comment: 8 page

Completely positive factorizations associated with Euclidean distance matrices corresponding to an arithmetic progression

Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties. We exploit those properties to develop completely positive factorizations of translations of those matrices. We show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix. We also discuss completely positive factorizations of such matrices over the integers. Methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties

Relation between non-exchangeability and measures of concordance of copulas

An investigation is presented of how a comprehensive choice of five most important measures of concordance (namely Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta, and their weaker counterpart Spearman's footrule) relate to non-exchangeability, i.e., asymmetry on copulas. Besides these results, the method proposed also seems to be new and may serve as a raw model for exploration of the relationship between a specific property of a copula and some of its measures of dependence structure, or perhaps the relationship between various measures of dependence structure themselves.Comment: 27 pages, 11 figure

Semitransitive and transitive subsemigroups of the inverse symmetric semigroups

We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups.Comment: 16 page

Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

We prove that the minimal cardinality of the semitransitive subsemigroup in the singular part \IS_n\setminus \S_n of the symmetric inverse semigroup \IS_n is $2n-p+1$, where $p$ is the greatest proper divisor of $n$, and classify all semitransitive subsemigroups of this minimal cardinality

Coherence and avoidance of sure loss for standardized functions and semicopulas

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.Comment: 31 pages, 2 figure