Narrow Results

Let ${ℛ}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(ℂ)$, the set of all $m\times n$ matrices over $ℂ$, to the set ${{ℛ}}^{†}$ of all Moore–Penrose invertible elements in ${ℛ}$. We study partial orders on ${{ℛ}}^{†}$ that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart ∗-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.

Fuzzy spatial models map a substantial degree of preference indifference. It has been shown that different definitions of covering result in different elements in the uncovered set when preference indifference is present. We consider several of the most frequently used definitions of covering relations found in the literature. The first definition that we examine yields an uncovered set, some of the elements of which are not Pareto efficient. Given that there is no reason to expect a set of players comprising a majority to settle for a Pareto deficient outcome, the remainder of the paper considers the ability of alternative definitions to avoid such a result.

We examine the effect of indifference on the existence of a majority rule maximal set. In our setting, it is shown in all but a limited number of cases that the maximal set is empty in an n-dimensional spatial model if and only if the Pareto set contains a union of cycles. The elements that constitute the exception are completely characterized.

Let $h:A\to A$ and $𝜀$ be a partial order on $A$. We deal with properties of oriented graphs which correspond to the algebra $(A,h)$ in the case that $h$ is monotone with respect to $𝜀$. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that $(A,h)$ is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that $h$ is monotone with respect to this order will be described; if the number of components of $(A,h)$ is infinite, then the number of such orders is equal to the cardinality of the power set of $A$.

In this paper, we derive coincidence point and common fixed point results under order homotopies of families of mappings in partially ordered $b$-metric spaces.

In this paper, we consider the set of all generalized inverses of an $m\times n$ matrix $A$, denoted by $A\{1\}$ and define two binary operations, ⊕ and ⋆ with some properties which offer the structure of a semi ring to $A\{1\}$. The compatibility of some partial orders like Sussman’s order, Conrad’s order, etc. is also checked which make it an ordered matrix semiring.

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order or a bucket order of all candidates, where bucket orders allow ties.

In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard.

The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in time for two voters. We show that it is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in , but not in NP or coNP unless NP = coNP, even for four voters.