I. IntroductionMotivated by a study of electrical network connections, Anderson and Duffin [1] introduced a binary operation A:B, called parallel addition, for pairs of positive (semi-definite) matrices. Subsequently Anderson and Trapp [3] have extended this notion to positive (linear) operators on a Hilbert space and demonstrated its importance in operator theory. On the other hand, Pusz and Woronowicz [12] considered a binary operation A # B, called geometric mean, for pairs of positive operators.In this paper, taking an axiomatic approach we introduce the notions of connection and mean. A connection a is a binary operation AaB for positive operators, which is monotone and continuous from above in each variable and satisfies the trans]brmer inequality, in the sense C(AaB) C <= (CA C) a(CBC).A connection a is a mean if it is normalized, i.e. lal = 1 where 1 denotes the identity operator. Such an axiomatic approach is already found in the paper of Nishio and Ando [11], in which parallel addition is given an axiomatic characterization. Addition, parallel addition and geometric mean are examples of connections. Though Bhagwat and Subramanian [6] introduced power means, for instance, {½(A p + BP)}l/P, these means are not monotone in general, nor satisfy the transformer inequality.The plan of the paper is as follows. In Sect. 2 the exact definition of connection is given. To each connection a there corresponds its transpose a', defined by (A,B)~BcrA. A natural order relation is introduced for pairs of connection. Section 3 is the principal part. The key is the existence of an affine orderisomorphism between the class of connections and the class of (positive) operatormonotone functions on (0, oe), those functions that are monotone of order n for every n in the sense of LSwner. This isomorphism a*--~f is characterized by the relation AaB = At/2f[A-1/2BA-1/2]A1/2

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Motivated by reachability questions in coherently controlled open quantum systems coupled to a thermal bath, as well as recent progress in the field of thermo-/vector-dmajorization [vom Ende and Dirr (2019)] we generalize classical majorization from unital quantum channels to channels with an arbitrary fixed point D of full rank. Such channels preserve some Gibbs-state and thus play an important role in the resource theory of quantum thermodynamics, in particular in Thermo-Majorization.Based on this we investigate D-majorization on matrices in terms of order properties, unique maximal and minimal elements, topological aspects, etc. Moreover we will characterize D-majorization in the qubit case and elaborate on why this is a challenging task when going beyond two dimensions.

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